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Higher integrability for anisotropic parabolic systems of p-Laplace type

Higher integrability for anisotropic parabolic systems of p-Laplace type 1IntroductionThe subject of this article are parabolic systems (1.1)ut−divDf(Du)=0inΩT=Ω×(0,T),{u}_{t}-{\rm{div}}Df\left(Du)=0\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{\Omega }_{T}=\Omega \times \left(0,T),where Ω⊂Rn(n≥2)\Omega \subset {{\mathbb{R}}}^{n}\hspace{0.33em}\left(n\ge 2)is a bounded domain (i.e., a nonempty, open and connected subset of Rn{{\mathbb{R}}}^{n}) and u:ΩT→RN(N≥1)u:{\Omega }_{T}\to {{\mathbb{R}}}^{N}\hspace{0.33em}\left(N\ge 1)is the desired function. By ut,Du{u}_{t},Du, we denote the derivative with respect to time respectively the spatial gradient of uu. We assume that f∈C2(RN×n)f\in {C}^{2}({{\mathbb{R}}}^{N\times n})satisfies a certain nonstandard growth condition. A very general class of equations with a nonstandard growth condition is given by the so-called pp–qqgrowth condition ∣z∣p≤f(ξ)≤L(1+∣ξ∣q){| z| }^{p}\le f\left(\xi )\le L\left(1+{| \xi | }^{q})with exponents 1<p<q1\lt p\lt q. A typical example for a function that fulfills such a condition is f(ξ)=1p∣ξ∣p+1q∣ξn∣qf\left(\xi )=\frac{1}{p}{| \xi | }^{p}+\frac{1}{q}{| {\xi }_{n}| }^{q}. In the elliptic setting, such equations, respectively the corresponding variational problem, min∫Ωf(Du)dx,\min \mathop{\int }\limits_{\Omega }f\left(Du){\rm{d}}x,were first examined in the late 1980s, starting with the seminal papers by Marcellini. In [19,25], Giaquinta and Marcellini gave counterexamples to the local Hölder continuity of minimizers (which was already known for the standard growth case p=qp=q), if the gap q−pq-pbetween the exponents is too large. Subsequently, for the scalar case N=1N=1, the local boundedness of the gradient of minimizers was proved by P. Marcellini under the condition that the exponents p,qp,qare not too far apart [26,27]. In the vectorial case N≥2N\ge 2, the partial Hölder continuity of weak solutions, i.e., Hölder continuity up to a subset with measure zero, was proved in [2]. There is a vast amount of literature on both elliptic and parabolic equations with pp–qqgrowth conditions, but we want to point out two recent articles in particular: In [9], the local boundedness of minimizers for functionals with anisotropic pp–qqgrowth conditions was established under sharp assumptions on the exponents, and [28] features a variational approach for a class of parabolic equations under very general growth conditions.In this article, however, we will not consider a general pp–qqgrowth condition but instead focus on a special case of anisotropic growth conditions. To make this precise, we assume that the integrand f∈C2(RN×n)f\in {C}^{2}({{\mathbb{R}}}^{N\times n})satisfies the following growth and ellipticity conditions: (1.2)∣f(ξ)∣≤L1+∑i=1n∣ξi∣pi∣D2f(ξ)∣≤L1+∑i=1n∣ξi∣pi−2⟨D2f(ξ)η,η⟩≥ν∑i=1n∣ξi∣pi−2∣ηi∣2\left\{\begin{array}{l}| f\left(\xi )| \le L\left(1+\mathop{\displaystyle \sum }\limits_{i=1}^{n}{| {\xi }_{i}| }^{{p}_{i}}\right)\hspace{1.0em}\\ | {D}^{2}f\left(\xi )| \le L\left(1+\mathop{\displaystyle \sum }\limits_{i=1}^{n}{| {\xi }_{i}| }^{{p}_{i}-2}\right)\hspace{1.0em}\\ \langle {D}^{2}f\left(\xi )\eta ,\eta \rangle \ge \nu \mathop{\displaystyle \sum }\limits_{i=1}^{n}{| {\xi }_{i}| }^{{p}_{i}-2}{| {\eta }_{i}| }^{2}\hspace{1.0em}\end{array}\right.for all ξ,η∈RN×n\xi ,\eta \in {{\mathbb{R}}}^{N\times n}with some constants 0<ν≤L0\lt \nu \le L. Here, we have denoted ξi=(ξi1,…,ξiN),ηi=(ηi1,…,ηiN)∈RN{\xi }_{i}=({\xi }_{i1},\ldots ,{\xi }_{iN}),{\eta }_{i}=({\eta }_{i1},\ldots ,{\eta }_{iN})\in {{\mathbb{R}}}^{N}. We also assume that the growth exponents p1,…,pn>1{p}_{1},\ldots ,{p}_{n}\gt 1satisfy min{pi}<max{pi}\min \{{p}_{i}\}\lt \max \{{p}_{i}\}. The prototype for such systems is the parabolic pi{p}_{i}-Laplace system ut−∑i=1n∂∂xi(∣Diu∣pi−2Diu)=0.{u}_{t}-\mathop{\sum }\limits_{i=1}^{n}\frac{\partial }{\partial {x}_{i}}({| {D}_{i}u| }^{{p}_{i}-2}{D}_{i}u)=0.This is precisely our system (1.1) for the choice f(ξ)=∑i=1n1pi∣ξi∣pi,f\left(\xi )=\mathop{\sum }\limits_{i=1}^{n}\frac{1}{{p}_{i}}{| {\xi }_{i}| }^{{p}_{i}},which obviously also satisfies the pp–qqgrowth condition with p=min{pi}p=\min \{{p}_{i}\}and q=max{pi}q=\max \{{p}_{i}\}. Without loss of generality, we can assume that the exponents are ordered, i.e., 1<p1≤p2≤…≤pn1\lt {p}_{1}\le {p}_{2}\le \ldots \le {p}_{n}, and hence pmin=p1,pmax=pn{p}_{\min }={p}_{1},{p}_{\max }={p}_{n}.Let us first mention a few important regularity results for elliptic equations satisfying the pi{p}_{i}-growth conditions (1.2), with no attempt at completeness. Similarly as for pp–qqgrowth conditions, the main feature of the theory is that the range of the exponents pi{p}_{i}must be sufficiently small for any regularity results to hold. The local boundedness of weak solutions under a sharp condition on the exponents pi{p}_{i}was proved in [18]. Based on this, the local Hölder continuity of bounded weak solutions in the special case p1=2{p}_{1}=2, p2=…=pn>2{p}_{2}=\ldots ={p}_{n}\gt 2was proved in [24]. This result was later extended to the case 1<p1<p2=…=pn1\lt {p}_{1}\lt {p}_{2}=\ldots ={p}_{n}in [13]. Recently, the local Hölder continuity of weak solutions was proved under the condition that pmax−pmin≤1c{p}_{\max }-{p}_{\min }\le \frac{1}{c}, with a constant ccthat depends only on the data [10]. In particular, this result allows all the exponents pi{p}_{i}to be different. In the superquadratic case 2≤p1≤…≤pn2\le {p}_{1}\hspace{0.33em}\le \ldots \le {p}_{n}, the local Lipschitz continuity of bounded weak solutions was proved in the very recent paper [4]. A sharp result about the speed of propagation for parabolic equations with pi{p}_{i}-growth can be found in [14].In this article, however, we will focus on the higher integrability of weak solutions in the parabolic setting. We want to show that the spatial derivatives Diu{D}_{i}u, which are a priori only in Lpi{L}^{{p}_{i}}, are in fact integrable to some higher power. In the elliptic setting, such higher integrability results for integrands with pp–qq-growth were proved in [15,16] and later refined in [8]. The elliptic result for the anisotropic growth conditions (1.2) can be found in [22]. In this article, we want to extend this result to the parabolic setting. For systems with a general pp–qq-growth condition, this has already been achieved in [6], where the higher integrability of the gradient was proved in the superquadratic case p≥2p\ge 2. The corresponding result for the subquadratic case p<2p\lt 2(and for an integrand ffwith (x,t)\left(x,t)-dependence) was proved in [31]. Furthermore, parabolic equations with pp–qq-growth and with (x,t)\left(x,t)-dependent coefficients were treated in [5]. Instead, we will only focus on the special growth conditions (1.2) in this article and restrict our attention to integrands ffthat depend only on the gradient variable. As will be discussed in Remark 1.5, in this special case, we can obtain a better bound on the gap between p1{p}_{1}and pn{p}_{n}than in the case of pp–qqgrowth conditions. Furthermore, we will only consider the superquadratic case, i.e., 2≤p1≤…≤pn2\le {p}_{1}\le \ldots \le {p}_{n}, and of course p1<pn{p}_{1}\lt {p}_{n}.To define weak solutions, we first need to introduce anisotropic versions of the classical Sobolev- and Bochner-spaces. Following Lions [23, Chapter 2.1.7], we denote Wxi1,pi(Ω,RN)={u∈Lpi(Ω,RN):Diu∈Lpi(Ω,RN)}{W}_{{x}_{i}}^{1,{p}_{i}}(\Omega ,{{\mathbb{R}}}^{N})=\{u\in {L}^{{p}_{i}}(\Omega ,{{\mathbb{R}}}^{N}):{D}_{i}u\in {L}^{{p}_{i}}(\Omega ,{{\mathbb{R}}}^{N})\}for a fixed index i∈{1,…,n}i\in \{1,\ldots ,n\}. Endowed with the norm ‖u‖Wxi1,pi≔‖u‖Lpi+‖Diu‖Lpi,{\Vert u\Vert }_{{W}_{{x}_{i}}^{1,{p}_{i}}}:= {\Vert u\Vert }_{{L}^{{p}_{i}}}+{\Vert {D}_{i}u\Vert }_{{L}^{{p}_{i}}},this space becomes a Banach space. Furthermore, we denote p=(p1,…,pn){\bf{p}}=\left({p}_{1},\ldots ,{p}_{n})and we define the anisotropic Sobolev space W1,p(Ω,RN){W}^{1,{\bf{p}}}(\Omega ,{{\mathbb{R}}}^{N})as follows: W1,p(Ω,RN)=⋂i=1nWxi1,pi(Ω,RN)={u∈Lpn(Ω,RN):Diu∈Lpi(Ω,RN)fori=1,…,n}.{W}^{1,{\bf{p}}}(\Omega ,{{\mathbb{R}}}^{N})=\mathop{\bigcap }\limits_{i=1}^{n}{W}_{{x}_{i}}^{1,{p}_{i}}(\Omega ,{{\mathbb{R}}}^{N})=\{u\in {L}^{{p}_{n}}(\Omega ,{{\mathbb{R}}}^{N}):{D}_{i}u\in {L}^{{p}_{i}}(\Omega ,{{\mathbb{R}}}^{N})\hspace{0.33em}\hspace{0.1em}\text{for}\hspace{0.1em}\hspace{0.33em}i=1,\ldots ,n\}.We equip this space with the norm ‖u‖W1,p(Ω,RN)≔‖u‖Lpn(Ω,RN)+∑i=1n‖Diu‖Lpi(Ω,RN).{\Vert u\Vert }_{{W}^{1,{\bf{p}}}\left(\Omega ,{{\mathbb{R}}}^{N})}:= {\Vert u\Vert }_{{L}^{{p}_{n}}\left(\Omega ,{{\mathbb{R}}}^{N})}+\mathop{\sum }\limits_{i=1}^{n}{\Vert {D}_{i}u\Vert }_{{L}^{{p}_{i}}\left(\Omega ,{{\mathbb{R}}}^{N})}.Now we will turn our attention to anisotropic Bochner spaces. For a fixed index i∈{1,…,n}i\in \{1,\ldots ,n\}, we denote by Lpi(0,T;Wxi1,pi(Ω,RN)){L}^{{p}_{i}}(0,T;\hspace{0.33em}{W}_{{x}_{i}}^{1,{p}_{i}}(\Omega ,{{\mathbb{R}}}^{N}))the “classical” Bochner space with respect to the variable xi{x}_{i}. A suitable anisotropic Bochner space, which is needed for the definition of weak solutions, is then defined in the following way.Definition 1.1The anisotropic Bochner space Lp(0,T;W1,p(Ω,RN)){L}^{{\bf{p}}}(0,T;\hspace{0.33em}{W}^{1,{\bf{p}}}(\Omega ,{{\mathbb{R}}}^{N}))is defined as follows: Lp(0,T;W1,p(Ω,RN))=⋂i=1nLpi(0,T;Wxi1,pi(Ω,RN)).{L}^{{\bf{p}}}(0,T;\hspace{0.33em}{W}^{1,{\bf{p}}}(\Omega ,{{\mathbb{R}}}^{N}))=\mathop{\bigcap }\limits_{i=1}^{n}{L}^{{p}_{i}}(0,T;\hspace{0.33em}{W}_{{x}_{i}}^{1,{p}_{i}}(\Omega ,{{\mathbb{R}}}^{N})).In particular, any function Lp(0,T;W1,p(Ω,RN)){L}^{{\bf{p}}}(0,T;\hspace{0.33em}{W}^{1,{\bf{p}}}(\Omega ,{{\mathbb{R}}}^{N}))has the following properties: (1)u(⋅,t)∈W1,p(Ω,RN)u\left(\cdot ,t)\in {W}^{1,{\bf{p}}}(\Omega ,{{\mathbb{R}}}^{N})for a.e. t∈(0,T)t\in \left(0,T)and the mapping (0,T)∋t↦u(⋅,t)∈W1,p(Ω,RN)\left(0,T)\hspace{0.33em}\ni \hspace{0.33em}t\mapsto u\left(\cdot ,t)\in {W}^{1,{\bf{p}}}(\Omega ,{{\mathbb{R}}}^{N})is strongly measurable.(2)The anisotropic Bochner norm ‖u‖≔∫0T‖u(⋅,t)‖Lpn(Ω,RN)pndt1pn+∑i=1n∫0T‖Diu(⋅,t)‖Lpi(Ω,RN)pidt1pi\Vert u\Vert := {\left(\underset{0}{\overset{T}{\int }}{\Vert u\left(\cdot ,t)\Vert }_{{L}^{{p}_{n}}\left(\Omega ,{{\mathbb{R}}}^{N})}^{{p}_{n}}{\rm{d}}t\right)}^{\tfrac{1}{{p}_{n}}}+\mathop{\sum }\limits_{i=1}^{n}{\left(\underset{0}{\overset{T}{\int }}{\Vert {D}_{i}u\left(\cdot ,t)\Vert }_{{L}^{{p}_{i}}\left(\Omega ,{{\mathbb{R}}}^{N})}^{{p}_{i}}{\rm{d}}t\right)}^{\tfrac{1}{{p}_{i}}}is finite.With this notion of anisotropic Bochner spaces at hand, we can define weak solutions for (1.1) in the following way.Definition 1.2A function u∈Lp(0,T;W1,p(Ω,RN))∩L∞(0,T;L2(Ω,RN))u\in {L}^{{\bf{p}}}(0,T;\hspace{0.33em}{W}^{1,{\bf{p}}}(\Omega ,{{\mathbb{R}}}^{N}))\cap {L}^{\infty }(0,T;\hspace{0.33em}{L}^{2}(\Omega ,{{\mathbb{R}}}^{N}))is called a weak solution of (1.1), if the weak formulation (1.3)∫ΩTu⋅φt−⟨Df(Du),Dφ⟩dz=0\mathop{\int }\limits_{{\Omega }_{T}}u\cdot {\varphi }_{t}-\langle Df\left(Du),D\varphi \rangle {\rm{d}}z=0holds for all test functions φ∈C0∞(ΩT,RN)\varphi \in {C}_{0}^{\infty }({\Omega }_{T},{{\mathbb{R}}}^{N}).Remark 1.3By “⋅<mml:mpadded xmlns:ali="http://www.niso.org/schemas/ali/1.0/"xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"voffset="-0.2ex">\cdot </mml:mpadded>”, we denote the scalar product on RN{{\mathbb{R}}}^{N}(or Rn{{\mathbb{R}}}^{n}), and by ⟨⋅,⋅⟩\langle \cdot ,\cdot \rangle , we denote the scalar product on RN×n{{\mathbb{R}}}^{N\times n}.The existence of weak solutions can be deduced from the theory of monotone operators, see [23, Chapter 2.1.7, Theorem 1.4]. In this article, the main focus lies instead on the (local) higher integrability of weak solutions, which can be used as a starting tool for further regularity results. Our main result is the following theorem.Theorem 1.4Let f∈C2(RN×n)f\in {C}^{2}({{\mathbb{R}}}^{N\times n})be a function satisfying the structure conditions (1.2). Furthermore, let us assume that the exponents pi{p}_{i}are ordered, i.e., p1≤p2≤…≤pn{p}_{1}\le {p}_{2}\le \ldots \le {p}_{n}and satisfy the gap condition(1.4)2≤p1<pn<p1+4n.2\le {p}_{1}\lt {p}_{n}\lt {p}_{1}+\frac{4}{n}.Let u be a weak solution of (1.1). Then we haveDu∈Llocs(ΩT,RN×n)∀s<p1+4n.Du\in {L}_{{\rm{loc}}}^{s}({\Omega }_{T},{{\mathbb{R}}}^{N\times n})\hspace{1.0em}\forall s\lt {p}_{1}+\frac{4}{n}.In other words, this theorem shows that DuDuis locally in Lpn+ε{L}^{{p}_{n}+\varepsilon }for some ε>0\varepsilon \gt 0. Of course, we can also use this result to improve the integrability of uu.Remark 1.5Let us compare the condition (1.4) with the corresponding elliptic condition from [22, Theorem A.1]. In the elliptic case, the exponents are required to satisfy the bound pn<np1n−2=p1+2p1n−2,{p}_{n}\lt \frac{n{p}_{1}}{n-2}={p}_{1}+\frac{2{p}_{1}}{n-2},while the parabolic bound can be rewritten as follows: pn<p1+4n=p1+2p1(n+2)−2⋅2p1.{p}_{n}\lt {p}_{1}+\frac{4}{n}={p}_{1}+\frac{2{p}_{1}}{\left(n+2)-2}\cdot \frac{2}{{p}_{1}}.This seems to be the natural bound since the dimension nnis replaced by n+2n+2(which is due to the typical parabolic scaling in time) and the parabolic deficit 2p1\frac{2}{{p}_{1}}shows up.We can also compare (1.4) to the corresponding condition for systems that satisfy a general pp–qq-growth condition. In the elliptic case, the higher integrability of DuDuwas proved under the condition that q−p<2pnq-p\lt \frac{2p}{n}[15, Theorem 2.1], while in the parabolic case, the gap condition is given by q−p<2pn+2⋅2p=4n+2q-p\lt \frac{2p}{n+2}\cdot \frac{2}{p}=\frac{4}{n+2}[6, Lemma 6.8], which is a more restrictive condition than (1.4). This is explained by the fact that for systems with a pp–qq-growth condition, the weak derivatives Diu{D}_{i}uare a priori only in Lp{L}^{p}. In our case, however, we know a priori that Diu{D}_{i}uis in Lpi{L}^{{p}_{i}}with pi≥p1{p}_{i}\ge {p}_{1}. This better integrability leads to a less restrictive condition on the gap pmax−pmin{p}_{\max }-{p}_{\min }.Remark 1.6For later use, we note that (1.4) implies (1.5)pn<p1+4n≤p1+2p1n=(n+2)p1n⇒pnp1<n+2n.{p}_{n}\lt {p}_{1}+\frac{4}{n}\le {p}_{1}+\frac{2{p}_{1}}{n}=\frac{\left(n+2){p}_{1}}{n}\hspace{0.33em}\Rightarrow \hspace{0.33em}\frac{{p}_{n}}{{p}_{1}}\lt \frac{n+2}{n}.We want to conclude this introduction with a brief outline of the proof of Theorem 1.4. The main idea, which was also used in [6,15,22], is to test the weak formulation (1.3) with a finite difference of uu, i.e., φ(x,t)≈u(x+hes,t)−u(x,t)\varphi \left(x,t)\approx u\left(x+h{e}_{s},t)-u\left(x,t). Via a Caccioppoli type inequality, we will then strive to obtain a uniform bound of the type (1.6)∫Qϱ(z0)∣Diu(x+hes,t)−Diu(x,t)∣pi∣h∣γpidz≤C,\mathop{\int }\limits_{{Q}_{\varrho }\left({z}_{0})}\frac{{| {D}_{i}u\left(x+h{e}_{s},t)-{D}_{i}u\left(x,t)| }^{{p}_{i}}}{{| h| }^{\gamma {p}_{i}}}{\rm{d}}z\le C,locally on any cylinder Qϱ(z0){Q}_{\varrho }\left({z}_{0}), where γ∈(0,1)\gamma \in \left(0,1)and Qϱ(z0)=Bϱ(x0)×(t0−ϱ2,t0){Q}_{\varrho }\left({z}_{0})={B}_{\varrho }\left({x}_{0})\times \left({t}_{0}-{\varrho }^{2},{t}_{0})denotes a standard parabolic cylinder with center z0=(x0,t0)∈ΩT{z}_{0}=\left({x}_{0},{t}_{0})\in {\Omega }_{T}and radius ϱ>0\varrho \gt 0. Roughly speaking, the estimate (1.6) asserts that the Lpi{L}^{{p}_{i}}-norm of a fractional difference quotient is uniformly bounded with respect to hh. From this, we can conclude that uubelongs to a certain fractional Sobolev space, and via an embedding theorem, we obtain a small amount of higher integrability for Diu{D}_{i}u. To obtain the full higher integrability, we then need to perform a suitable iteration procedure in the last step.The article is organized in the following way: In Section 2, we gather some preliminaries, which will be needed later on. In Section 3, we prove the Caccioppoli type inequality for finite differences. The iteration procedure, that is needed for the proof of Theorem 1.4, will be performed in the last section.2PreliminariesIn this chapter, we gather some important tools that will be needed for the proof of the higher integrability, in particular, concerning fractional Sobolev spaces and difference quotients respectively finite differences.2.1Some useful inequalitiesTo absorb certain terms, we will need the following standard iteration lemma [21, Lemma 6.1].Lemma 2.1Let ϑ∈(0,1){\vartheta }\in \left(0,1), A,B≥0A,B\ge 0, and α>0\alpha \gt 0. There exists a constant C(α,ϑ)C\left(\alpha ,{\vartheta })such that there holds: For any r∈(0,ϱ)r\in \left(0,\varrho )and any nonnegative, bounded function Φ:[r,ϱ]→[0,∞)\Phi :\left[r,\varrho ]\to \left[0,\infty )satisfyingΦ(s)≤ϑΦ(t)+A(t−s)α+B∀r≤s<t≤ϱ,\Phi \left(s)\le {\vartheta }\Phi \left(t)+\frac{A}{{\left(t-s)}^{\alpha }}+B\hspace{1.0em}\forall r\le s\lt t\le \varrho ,we haveΦ(r)≤CA(ϱ−r)α+B.\Phi \left(r)\le C\left[\frac{A}{{\left(\varrho -r)}^{\alpha }}+B\right].For the proof of the Caccioppoli inequality, we will need the following technical lemma. The case σ<0\sigma \lt 0was proved in [1, Lemma 2.1] and the case σ≥0\sigma \ge 0in [20, Lemma 2.1].Lemma 2.2Let k∈Nk\in {\mathbb{N}}. For every σ>−12\sigma \gt -\frac{1}{2}, there exists a constant C=C(σ)≥1C=C\left(\sigma )\ge 1, such that the following estimate holds: 1C(μ2+∣A∣2+∣B∣2)σ≤∫01(μ2+∣A+s(B−A)∣2)σds≤C(μ2+∣A∣2+∣B∣2)σ\frac{1}{C}{({\mu }^{2}+{| A| }^{2}+{| B| }^{2})}^{\sigma }\le \underset{0}{\overset{1}{\int }}{({\mu }^{2}+{| A+s\left(B-A)| }^{2})}^{\sigma }{\rm{d}}s\le C{({\mu }^{2}+{| A| }^{2}+{| B| }^{2})}^{\sigma }for any μ≥0\mu \ge 0and any A,B∈RkA,B\in {{\mathbb{R}}}^{k}, not both zero if μ=0\mu =0and σ<0\sigma \lt 0.The following lemma [30, Lemma 3.2] is a parabolic version of the Sobolev inequality and follows from the Gagliardo-Nirenberg inequality.Lemma 2.3Let σ≥1\sigma \ge 1, Qϱ(z0)⊂ΩT{Q}_{\varrho }\left({z}_{0})\subset {\Omega }_{T}andu∈Lσ(t0−ϱ2,t0;W1,σ(Bϱ(x0),RN))∩L∞(t0−ϱ2,t0;L2(Bϱ(x0),RN)).u\in {L}^{\sigma }({t}_{0}-{\varrho }^{2},{t}_{0};\hspace{0.33em}{W}^{1,\sigma }({B}_{\varrho }\left({x}_{0}),{{\mathbb{R}}}^{N}))\cap {L}^{\infty }({t}_{0}-{\varrho }^{2},{t}_{0};\hspace{0.33em}{L}^{2}({B}_{\varrho }\left({x}_{0}),{{\mathbb{R}}}^{N})).There exists a constant C=C(N,n,σ)C=C\left(N,n,\sigma ), such that for any radius ϱ2≤r<ϱ\frac{\varrho }{2}\le r\lt \varrho , the following estimate holds: ∫Qr(z0)∣u∣σ(n+2)ndz≤C∫Qϱ(z0)∣Du∣σ+uϱ−rσdzsupt∈(t0−ϱ2,t0)∫Bϱ(x0)∣u(⋅,t)∣2dxσn.\mathop{\int }\limits_{{Q}_{r}\left({z}_{0})}{| u| }^{\tfrac{\sigma \left(n+2)}{n}}{\rm{d}}z\le C\mathop{\int }\limits_{{Q}_{\varrho }\left({z}_{0})}\left({| Du| }^{\sigma }+{\left|\frac{u}{\varrho -r}\right|}^{\sigma }\right){\rm{d}}z{\left(\mathop{\sup }\limits_{t\in \left({t}_{0}-{\varrho }^{2},{t}_{0})}\mathop{\int }\limits_{{B}_{\varrho }\left({x}_{0})}{| u\left(\cdot ,t)| }^{2}{\rm{d}}x\right)}^{\tfrac{\sigma }{n}}.2.2Finite differences and fractional Sobolev spacesLet v:ΩT→RNv:{\Omega }_{T}\to {{\mathbb{R}}}^{N}be some function. By τs,hv:ΩTh→RN{\tau }_{s,h}v:{\Omega }_{T}^{h}\to {{\mathbb{R}}}^{N}, we denote the finite difference of vvin the spatial direction s∈{1,…,n}s\in \{1,\ldots ,n\}with increment h∈Rh\in {\mathbb{R}}, i.e., τs,hv(x,t)=v(x+hes,t)−v(x,t).{\tau }_{s,h}v\left(x,t)=v\left(x+h{e}_{s},t)-v\left(x,t).By ΩTh={x∈Ω:dist(x,∂Ω)>∣h∣}×(0,T){\Omega }_{T}^{h}=\{x\in \Omega :{\rm{dist}}\left(x,\partial \Omega )\gt | h| \}\times \left(0,T), we have denoted the inner parallel cylinder (with respect to space) at distance ∣h∣| h| . We will need the following simple estimate.Lemma 2.4Let Q2R(z0)⊂ΩT{Q}_{2R}\left({z}_{0})\subset {\Omega }_{T}, p∈[1,∞)p\in \left[1,\infty ), and f∈Lp(Q2R(z0))f\in {L}^{p}\left({Q}_{2R}\left({z}_{0})). There exists a positive constant C=C(p)C=C\left(p), such that for any h∈(−R,R)h\in \left(-R,R), the following inequality holds: (2.1)∫QR(1+∣f∣+∣τs,hf∣)pdz≤C∫Q2R(1+∣f∣)pdz.\mathop{\int }\limits_{{Q}_{R}}{\left(1+| f| +| {\tau }_{s,h}f| )}^{p}{\rm{d}}z\le C\mathop{\int }\limits_{{Q}_{2R}}{\left(1+| f| )}^{p}{\rm{d}}z.The next lemma, which can be found, e.g., in [17, Chapter 5.8.2, Theorem 3], asserts that the Lp{L}^{p}-norm of a difference quotient on some cylinder is bounded by the Lp{L}^{p}-norm of the respective partial derivative on a larger cylinder.Lemma 2.5Let f,Dsf∈Lp(Q2R)f,{D}_{s}f\in {L}^{p}\left({Q}_{2R})with s∈{1,…,n}s\in \{1,\ldots ,n\}and p∈[1,∞)p\in \left[1,\infty ). Then, for any h∈(−R,R)h\in \left(-R,R), the following inequality holds: (2.2)∫QR∣τs,hf∣pdz≤∣h∣p∫Q2R∣Dsf∣pdz.\mathop{\int }\limits_{{Q}_{R}}{| {\tau }_{s,h}f| }^{p}{\rm{d}}z\le {| h| }^{p}\mathop{\int }\limits_{{Q}_{2R}}{| {D}_{s}f| }^{p}{\rm{d}}z.Remark 2.6Obviously, the last inequality can be rewritten as follows: ∫QRf(x+hes,t)−f(x,t)hpdz≤∫Q2R∣Dsf∣pdz,\mathop{\int }\limits_{{Q}_{R}}{\left|\frac{f\left(x+h{e}_{s},t)-f\left(x,t)}{h}\right|}^{p}{\rm{d}}z\le \mathop{\int }\limits_{{Q}_{2R}}{| {D}_{s}f| }^{p}{\rm{d}}z,i.e., the Lp{L}^{p}-norm of the difference quotient in direction ss(on a parabolic cylinder) is bounded by the Lp{L}^{p}-norm of the partial derivative Dsf{D}_{s}fon a larger cylinder, and this bound is uniform with respect to hh.Later on, we will use these finite differences to show that the partial derivatives of a weak solution belong to some fractional Sobolev space, from which the higher integrability follows via an embedding theorem. Thus, we also need a few results about elliptic and parabolic fractional Sobolev spaces. Let us first recall the definition of a fractional Sobolev space in the elliptic setting [11, Section 2]. Let k∈N0k\in {{\mathbb{N}}}_{0}and p≥1p\ge 1. We say that a function f∈Wk,p(Ω,RN)f\in {W}^{k,p}(\Omega ,{{\mathbb{R}}}^{N})belongs to the fractional Sobolev space (or Sobolev-Slobodeckij space) Wk+α,p(Ω,RN){W}^{k+\alpha ,p}(\Omega ,{{\mathbb{R}}}^{N}), for some α∈(0,1)\alpha \in \left(0,1), if the Gagliardo semi-norm [Dβf]α,p;Ωp≔∫Ω∫Ω∣Dβf(x)−Dβf(y)∣p∣x−y∣n+αpdxdy{[}{D}^{\beta }f{]}_{\alpha ,p;\hspace{0.33em}\Omega }^{p}:= \mathop{\int }\limits_{\Omega }\mathop{\int }\limits_{\Omega }\frac{| {D}^{\beta }f\left(x)-{D}^{\beta }f(y){| }^{p}}{{| x-y| }^{n+\alpha p}}{\rm{d}}x{\rm{d}}yis finite for any multiindex β∈N0n\beta \in {{\mathbb{N}}}_{0}^{n}with ∣β∣=k| \beta | =k. The space Wk+α,p(Ω,RN){W}^{k+\alpha ,p}(\Omega ,{{\mathbb{R}}}^{N}), endowed with the norm ‖f‖Wk+α,p(Ω)≔‖f‖Wk,p(Ω)+∑∣β∣=k[Dβf]α,p;Ω,{\Vert f\Vert }_{{W}^{k+\alpha ,p}\left(\Omega )}:= {\Vert f\Vert }_{{W}^{k,p}\left(\Omega )}+\sum _{| \beta | =k}{[}{D}^{\beta }f{]}_{\alpha ,p;\Omega },is a Banach space. For later use, we state the following interpolation result for fractional Sobolev spaces, which is essentially an anisotropic version of the interpolation result from [7, Corollary 3.2]. The proof can be found in the appendix.Lemma 2.7Let λ,μ∈(0,1)\lambda ,\mu \in \left(0,1)and p∈(1,∞)p\in \left(1,\infty ). Let α∈(0,1)\alpha \in \left(0,1)be such that1+α=θ(1+λ)+(1−θ)μ1+\alpha =\theta \left(1+\lambda )+\left(1-\theta )\mu for some θ∈(0,1)\theta \in \left(0,1). Furthermore, let1r=θp+1−θ2.\frac{1}{r}=\frac{\theta }{p}+\frac{1-\theta }{2}.Finally, let f∈W1,p(Rn)∩Wμ,2(Rn)f\in {W}^{1,p}\left({{\mathbb{R}}}^{n})\cap {W}^{\mu ,2}\left({{\mathbb{R}}}^{n})and D1f∈Wλ,p(Rn){D}_{1}f\in {W}^{\lambda ,p}\left({{\mathbb{R}}}^{n}). Then we have D1f∈Wα,r(Rn){D}_{1}f\in {W}^{\alpha ,r}\left({{\mathbb{R}}}^{n}), and the estimate(2.3)‖D1f‖Wα,r≤C[‖D1f‖Lp+[D1f]λ,p]θ‖f‖Wμ,21−θ{\Vert {D}_{1}f\Vert }_{{W}^{\alpha ,r}}\le C{{[}{\Vert {D}_{1}f\Vert }_{{L}^{p}}+{\left[{D}_{1}f]}_{\lambda ,p}]}^{\theta }{\Vert f\Vert }_{{W}^{\mu ,2}}^{1-\theta }holds.Remark 2.8The estimate (2.3) also holds for all α∈(0,1)\alpha \in \left(0,1)such that 1+α≤θ(1+λ)+(1−θ)μ.1+\alpha \le \theta \left(1+\lambda )+\left(1-\theta )\mu .This follows from the continuous embedding Wα,r(Rn)↪Wα′,r(Rn){W}^{\alpha ,r}\left({{\mathbb{R}}}^{n})\hspace{0.33em}\hookrightarrow \hspace{0.33em}{W}^{\alpha ^{\prime} ,r}\left({{\mathbb{R}}}^{n})for 0<α′≤α<10\lt \alpha ^{\prime} \le \alpha \lt 1, see, e.g., [11, Proposition 2.1].We also recall the following classical embedding theorem, which can be found in [11, Theorem 6.5].Lemma 2.9Let α∈(0,1)\alpha \in \left(0,1)and p∈[1,∞)p\in \left[1,\infty )such that αp<n\alpha p\lt n. There exists a constant C(n,p,α)C\left(n,p,\alpha )such that for any f∈Wα,p(Rn)f\in {W}^{\alpha ,p}\left({{\mathbb{R}}}^{n}), we have‖f‖Lnpn−αp(Rn)≤C[f]α,p;Rn.{\Vert f\Vert }_{{L}^{\tfrac{np}{n-\alpha p}}\left({{\mathbb{R}}}^{n})}\le C\hspace{0.33em}{[f]}_{\alpha ,p;{{\mathbb{R}}}^{n}}.Hence, the space Wα,p(Rn){W}^{\alpha ,p}\left({{\mathbb{R}}}^{n})is continuously embedded into Lnpn−αp(Rn){L}^{\tfrac{np}{n-\alpha p}}\left({{\mathbb{R}}}^{n}).With these prerequisites at hand, we can prove the following fractional version of the Gagliardo-Nirenberg inequality.Lemma 2.10Let Bϱ(x0)⊂Rn{B}_{\varrho }\left({x}_{0})\subset {{\mathbb{R}}}^{n}be a ball and let λ,μ,θ∈(0,1)\lambda ,\mu ,\theta \in \left(0,1), 1<p,2<s<∞1\lt p,2\lt s\lt \infty , such that(2.4)−ns≤θλ−np−(1−θ)1−μ+n2.-\frac{n}{s}\le \theta \left(\lambda -\frac{n}{p}\right)-\left(1-\theta )\left(1-\mu +\frac{n}{2}\right).Suppose that f∈W1,p(Bϱ(x0))∩Wμ,2(Bϱ(x0))f\in {W}^{1,p}\left({B}_{\varrho }\left({x}_{0}))\cap {W}^{\mu ,2}\left({B}_{\varrho }\left({x}_{0}))and Dif∈Wλ,p(Bϱ(x0)){D}_{i}f\in {W}^{\lambda ,p}\left({B}_{\varrho }\left({x}_{0}))for a fixed index i∈{1,…,n}i\in \{1,\ldots ,n\}. Then Dif∈Ls(Br(x0)){D}_{i}f\in {L}^{s}\left({B}_{r}\left({x}_{0}))for any radius r∈(0,ϱ)r\in \left(0,\varrho ), and there exists a constant C(n,μ,λ,p,s,θ,1/(ϱ−r))C\left(n,\mu ,\lambda ,p,s,\theta ,1\hspace{0.1em}\text{/}\hspace{0.1em}\left(\varrho -r))such that‖Dif‖Ls(Br(x0))≤C(‖f‖W1,p(Bϱ(x0))+[Dif]λ,p;Bϱ(x0))θ‖f‖Wμ,2(Bϱ(x0))1−θ.{\Vert {D}_{i}f\Vert }_{{L}^{s}\left({B}_{r}\left({x}_{0}))}\le C{({\Vert f\Vert }_{{W}^{1,p}\left({B}_{\varrho }\left({x}_{0}))}+{{[}{D}_{i}f]}_{\lambda ,p;{B}_{\varrho }\left({x}_{0})})}^{\theta }{\Vert f\Vert }_{{W}^{\mu ,2}\left({B}_{\varrho }\left({x}_{0}))}^{1-\theta }.The aforementioned lemma is a slightly modified version of [6, Lemma 6.4]. The difference lies in the fact that we do not obtain the higher integrability of DfDf, but only of a fixed partial derivative Dif{D}_{i}f. To obtain the higher integrability of Dif{D}_{i}f, it is sufficient to presuppose that Dif∈Wλ,p{D}_{i}f\in {W}^{\lambda ,p}, which is a weaker condition than f∈W1+λ,pf\in {W}^{1+\lambda ,p}. As we will see later, this is a necessary adaptation to the anisotropic setting. The proof is very similar to [6], but we include it for completeness.ProofTo simplify the notation, we suppress the center of the ball and write Bϱ{B}_{\varrho }instead of Bϱ(x0){B}_{\varrho }\left({x}_{0}). Let η∈C0∞(Bϱ)\eta \in {C}_{0}^{\infty }\left({B}_{\varrho })be a cutoff function such that 0≤η≤10\le \eta \le 1, η=1\eta =1on Br{B}_{r}and ∣Dη∣≤2ϱ−r| D\eta | \le \frac{2}{\varrho -r}. Now we choose the parameter α\alpha such that (2.5)n+αsns=θp+1−θ2⇔α=θnp+(1−θ)n2−ns.\frac{n+\alpha s}{ns}=\frac{\theta }{p}+\frac{1-\theta }{2}\hspace{0.33em}\iff \hspace{0.33em}\alpha =\frac{\theta n}{p}+\frac{\left(1-\theta )n}{2}-\frac{n}{s}.Due to (2.4), we have (2.6)α≤θnp+(1−θ)n2+θλ−np−(1−θ)1−μ+n2=θλ−(1−θ)(1−μ)<1.\alpha \le \frac{\theta n}{p}+\frac{\left(1-\theta )n}{2}+\theta \left(\lambda -\frac{n}{p}\right)-\left(1-\theta )\left(1-\mu +\frac{n}{2}\right)=\theta \lambda -\left(1-\theta )\left(1-\mu )\lt 1.Since p,2<sp,2\lt s, we also have α=θnp−ns+(1−θ)n2−ns>0,\alpha =\theta \left(\frac{n}{p}-\frac{n}{s}\right)+\left(1-\theta )\left(\frac{n}{2}-\frac{n}{s}\right)\gt 0,and hence, α∈(0,1)\alpha \in \left(0,1). Therefore, we can apply Lemma 2.9 with α,nsn+αs\left(\alpha ,\frac{ns}{n+\alpha s}\right)instead of (α,p)\left(\alpha ,p)to infer that ‖Dif‖Ls(Br)≤‖Di(fη)‖Ls(Rn)≤C(n,s,α)[Di(fη)]α,nsn+αs;Rn.{\Vert {D}_{i}f\Vert }_{{L}^{s}\left({B}_{r})}\le {\Vert {D}_{i}(f\eta )\Vert }_{{L}^{s}\left({{\mathbb{R}}}^{n})}\le C\left(n,s,\alpha ){{[}{D}_{i}(f\eta )]}_{\alpha ,\tfrac{ns}{n+\alpha s};{{\mathbb{R}}}^{n}}.Due to the upper bound (2.6), which is a consequence of the choice of α\alpha from (2.5), we have 1+α≤θ(1+λ)+(1−θ)μ1+\alpha \le \theta \left(1+\lambda )+\left(1-\theta )\mu . Hence, we can apply Lemma 2.7 resp. Remark 2.8 with nsn+αs\frac{ns}{n+\alpha s}instead of rrto the right-hand side. This yields [Di(fη)]α,nsn+αs;Rn≤C(‖Di(fη)‖Lp(Rn)+[Di(fη)]λ,p;Rn)θ‖fη‖Wμ,2(Rn)1−θ≤C(‖Di(fη)‖Lp(Bϱ)+[Di(fη)]λ,p;Bϱ)θ‖f‖Wμ,2(Bϱ)1−θ.\begin{array}{rcl}{{[}{D}_{i}(f\eta )]}_{\alpha ,\tfrac{ns}{n+\alpha s};{{\mathbb{R}}}^{n}}& \le & C{({\Vert {D}_{i}(f\eta )\Vert }_{{L}^{p}\left({{\mathbb{R}}}^{n})}+{{[}{D}_{i}(f\eta )]}_{\lambda ,p;{{\mathbb{R}}}^{n}})}^{\theta }{\Vert f\eta \Vert }_{{W}^{\mu ,2}\left({{\mathbb{R}}}^{n})}^{1-\theta }\\ & \le & C{({\Vert {D}_{i}(f\eta )\Vert }_{{L}^{p}\left({B}_{\varrho })}+{{[}{D}_{i}(f\eta )]}_{\lambda ,p;{B}_{\varrho }})}^{\theta }{\Vert f\Vert }_{{W}^{\mu ,2}\left({B}_{\varrho })}^{1-\theta }.\end{array}In this estimate, the constant CCdepends on λ,μ,p\lambda ,\mu ,p, and θ\theta . Now we use the fact that ∣η∣≤1,∣Dη∣≤2ϱ−r| \eta | \le 1,| D\eta | \le \frac{2}{\varrho -r}to estimate the Gagliardo-Seminorm on the right-hand side in the following way: [Di(fη)]λ,p;Bϱ=∫Bϱ∫Bϱ∣Dif(x)η(x)−Dif(y)η(y)+f(x)Diη(x)−f(y)Diη(y)∣p∣x−y∣n+λpdxdy1p≤Cp,1ϱ−r[Dif]λ,p;Bϱ+∫Bϱ∫Bϱ∣f(x)−f(y)∣p∣x−y∣n+λpdxdy1p=Cp,1ϱ−r[[Dif]λ,p;Bϱ+[f]λ,p;Bϱ]≤Cp,1ϱ−r[‖f‖W1,p(Bϱ)+[Dif]λ,p;Bϱ].\begin{array}{rcl}{\left[{D}_{i}(f\eta )]}_{\lambda ,p;{B}_{\varrho }}& =& {\left(\mathop{\displaystyle \int }\limits_{{B}_{\varrho }}\mathop{\displaystyle \int }\limits_{{B}_{\varrho }}\frac{{| {D}_{i}f\left(x)\eta \left(x)-{D}_{i}f(y)\eta (y)+f\left(x){D}_{i}\eta \left(x)-f(y){D}_{i}\eta (y)| }^{p}}{{| x-y| }^{n+\lambda p}}{\rm{d}}x{\rm{d}}y\right)}^{\tfrac{1}{p}}\\ & \le & C\left(p,\frac{1}{\varrho -r}\right)\left[{{[}{D}_{i}f]}_{\lambda ,p;{B}_{\varrho }}+{\left(\mathop{\displaystyle \int }\limits_{{B}_{\varrho }}\mathop{\displaystyle \int }\limits_{{B}_{\varrho }}\frac{{| f\left(x)-f(y)| }^{p}}{{| x-y| }^{n+\lambda p}}{\rm{d}}x{\rm{d}}y\right)}^{\tfrac{1}{p}}\right]\\ & =& C\left(p,\frac{1}{\varrho -r}\right){[}{{[}{D}_{i}f]}_{\lambda ,p;{B}_{\varrho }}+{{[}f]}_{\lambda ,p;{B}_{\varrho }}]\\ & \le & C\left(p,\frac{1}{\varrho -r}\right){[}{\Vert f\Vert }_{{W}^{1,p}\left({B}_{\varrho })}+{{[}{D}_{i}f]}_{\lambda ,p;{B}_{\varrho }}].\end{array}Similarly, we also have ‖Di(fη)‖Lp(Bϱ)≤Cp,1ϱ−r‖f‖W1,p(Bϱ){\Vert {D}_{i}(f\eta )\Vert }_{{L}^{p}\left({B}_{\varrho })}\le C\left(p,\frac{1}{\varrho -r}\right){\Vert f\Vert }_{{W}^{1,p}\left({B}_{\varrho })}. Hence, we obtain the desired estimate ‖Dif‖Ls(Br)≤C(‖f‖W1,p(Bϱ)+[Dif]λ,p;Bϱ)θ‖f‖Wμ,2(Bϱ)1−θ,{\Vert {D}_{i}f\Vert }_{{L}^{s}\left({B}_{r})}\le C{({\Vert f\Vert }_{{W}^{1,p}\left({B}_{\varrho })}+{{[}{D}_{i}f]}_{\lambda ,p;{B}_{\varrho }})}^{\theta }{\Vert f\Vert }_{{W}^{\mu ,2}\left({B}_{\varrho })}^{1-\theta },with a constant that depends on n,s,α,λ,μ,p,θn,s,\alpha ,\lambda ,\mu ,p,\theta , and 1ϱ−r\frac{1}{\varrho -r}. Since α\alpha itself depends on θ,n,p\theta ,n,p, and ss(see (2.5)), we ultimately have C=C(n,μ,λ,p,s,θ,1/(ϱ−r))C=C\left(n,\mu ,\lambda ,p,s,\theta ,1\hspace{0.1em}\text{/}\hspace{0.1em}\left(\varrho -r)).□Of course, we will also need a parabolic version of such fractional Sobolev spaces. Let k∈N0k\in {{\mathbb{N}}}_{0}, p≥1p\ge 1and α∈(0,1)\alpha \in \left(0,1). A function u∈Lp(0,T;Wk,p(Ω,RN))u\in {L}^{p}(0,T;\hspace{0.33em}{W}^{k,p}(\Omega ,{{\mathbb{R}}}^{N}))belongs to the parabolic fractional Sobolev space Lp(0,T;Wk+α,p(Ω,RN)){L}^{p}(0,T;\hspace{0.33em}{W}^{k+\alpha ,p}(\Omega ,{{\mathbb{R}}}^{N})), if the parabolic Gagliardo semi-norm [Dβu]α,0,p;ΩTp≔∫0T∫Ω∫Ω∣Dβu(x,t)−Dβu(y,t)∣p∣x−y∣n+αpdxdydt{[}{D}^{\beta }u{]}_{\alpha ,0,p;\hspace{0.33em}{\Omega }_{T}}^{p}:= \underset{0}{\overset{T}{\int }}\mathop{\int }\limits_{\Omega }\mathop{\int }\limits_{\Omega }\frac{| {D}^{\beta }u\left(x,t)-{D}^{\beta }u(y,t){| }^{p}}{{| x-y| }^{n+\alpha p}}{\rm{d}}x{\rm{d}}y{\rm{d}}tis finite for any β∈N0n\beta \in {{\mathbb{N}}}_{0}^{n}with ∣β∣=k| \beta | =k. Similarly as in the elliptic case, the space Lp(0,T;Wk+α,p(Ω,RN)){L}^{p}(0,T;\hspace{0.33em}{W}^{k+\alpha ,p}(\Omega ,{{\mathbb{R}}}^{N})), endowed with the norm ‖u‖k+α,0,p;ΩT≔‖u‖Lp(0,T;Wk,p(Ω,RN))+∑∣β∣=k[Dβf]α,0,p;ΩT,{\Vert u\Vert }_{k+\alpha ,0,p;{\Omega }_{T}}:= {\Vert u\Vert }_{{L}^{p}\left(0,T;{W}^{k,p}\left(\Omega ,{{\mathbb{R}}}^{N}))}+\sum _{| \beta | =k}{[}{D}^{\beta }f{]}_{\alpha ,0,p;{\Omega }_{T}},is a Banach space. The following lemma is an anisotropic version of the parabolic fractional Sobolev inequality from [6, Lemma 6.5].Lemma 2.11Let Qϱ(z0)⊂ΩT{Q}_{\varrho }\left({z}_{0})\subset {\Omega }_{T}be a parabolic cylinder with radius ϱ≤1\varrho \le 1. Let λ,μ∈(0,1)\lambda ,\mu \in \left(0,1), 1<p,2<s<∞1\lt p,2\lt s\lt \infty be parameters such that(2.7)(s−p)1−μ+n2≤λp.\left(s-p)\left(1-\mu +\frac{n}{2}\right)\le \lambda p.Furthermore, let us assume thatu∈Lp(t0−ϱ2,t0;W1,p(Bϱ(x0)))∩L∞(t0−ϱ2,t0;Wμ,2(Bϱ(x0)))u\in {L}^{p}\left({t}_{0}-{\varrho }^{2},{t}_{0};\hspace{0.33em}{W}^{1,p}\left({B}_{\varrho }\left({x}_{0})))\cap {L}^{\infty }\left({t}_{0}-{\varrho }^{2},{t}_{0};\hspace{0.33em}{W}^{\mu ,2}\left({B}_{\varrho }\left({x}_{0})))andDiu∈Lp(t0−ϱ2,t0;Wλ,p(Bϱ(x0))){D}_{i}u\in {L}^{p}\left({t}_{0}-{\varrho }^{2},{t}_{0};\hspace{0.33em}{W}^{\lambda ,p}\left({B}_{\varrho }\left({x}_{0})))for a fixed index i∈{1,…,n}i\in \{1,\ldots ,n\}. Then we have Diu∈Ls(Br(x0)×(t0−ϱ2,t0)){D}_{i}u\in {L}^{s}\left({B}_{r}\left({x}_{0})\times \left({t}_{0}-{\varrho }^{2},{t}_{0}))for any radius r∈(0,ϱ)r\in \left(0,\varrho ), and there exists a constant C(n,μ,λ,p,s,1/(ϱ−r))C\left(n,\mu ,\lambda ,p,s,1\hspace{0.1em}\text{/}\hspace{0.1em}\left(\varrho -r))such that the following estimate holds: ‖Diu‖Ls(Br(x0)×(t1,t0))≤C(‖u‖Lp(t1,t0;W1,p(Bϱ))+[Diu]λ,0,p;Bϱ×(t1,t0))pssupt∈(t1,t0)‖u(⋅,t)‖Wμ,2(Bϱ)s−ps,\Vert {D}_{i}u{\Vert }_{{L}^{s}\left({B}_{r}\left({x}_{0})\times \left({t}_{1},{t}_{0}))}\le C{({\Vert u\Vert }_{{L}^{p}\left({t}_{1},{t}_{0};{W}^{1,p}\left({B}_{\varrho }))}+{\left[{D}_{i}u]}_{\lambda ,0,p;{B}_{\varrho }\times \left({t}_{1},{t}_{0})})}^{\tfrac{p}{s}}\mathop{\sup }\limits_{t\in \left({t}_{1},{t}_{0})}{\Vert u\left(\cdot ,t)\Vert }_{{W}^{\mu ,2}\left({B}_{\varrho })}^{\frac{s-p}{s}},where we used the abbreviations t1≔t0−ϱ2{t}_{1}:= {t}_{0}-{\varrho }^{2}and Bϱ≔Bϱ(x0){B}_{\varrho }:= {B}_{\varrho }\left({x}_{0}).ProofTo simplify the notation, we suppress the center of the ball and write Bϱ{B}_{\varrho }instead of Bϱ(x0){B}_{\varrho }\left({x}_{0}). For almost every time-slice t∈(t1,t0)t\in \left({t}_{1},{t}_{0}), we have u(⋅,t)∈W1,p(Bϱ)∩Wμ,2(Bϱ)u\left(\cdot ,t)\in {W}^{1,p}\left({B}_{\varrho })\cap {W}^{\mu ,2}\left({B}_{\varrho })and Diu(⋅,t)∈Wλ,p(Bϱ){D}_{i}u\left(\cdot ,t)\in {W}^{\lambda ,p}\left({B}_{\varrho }). Hence, we can apply Lemma 2.10 with θ=ps\theta =\frac{p}{s}. We note that the condition (2.4) is satisfied due to (2.7). We obtain ‖Diu(⋅,t)‖Ls(Br)≤C(‖u(⋅,t)‖W1,p(Bϱ)+[Diu(⋅,t)]λ,p;Bϱ)ps‖u(⋅,t)‖Wμ,2(Bϱ)s−ps{\Vert {D}_{i}u\left(\cdot ,t)\Vert }_{{L}^{s}\left({B}_{r})}\le C{({\Vert u\left(\cdot ,t)\Vert }_{{W}^{1,p}\left({B}_{\varrho })}+{{[}{D}_{i}u\left(\cdot ,t)]}_{\lambda ,p;{B}_{\varrho }})}^{\tfrac{p}{s}}{\Vert u\left(\cdot ,t)\Vert }_{{W}^{\mu ,2}\left({B}_{\varrho })}^{\frac{s-p}{s}}for almost every time-slice t∈(t1,t0)t\in \left({t}_{1},{t}_{0}). We integrate this inequality with respect to time to obtain ∫t1t0∫Br∣Diu∣sdxdt≤C∫t1t0(‖u(⋅,t)‖W1,p(Bϱ)+[Diu(⋅,t)]λ,p;Bϱ)p‖u(⋅,t)‖Wμ,2(Bϱ)s−pdt≤C∫t1t0(‖u(⋅,t)‖W1,p(Bϱ)+[Diu(⋅,t)]λ,p;Bϱ)pdtsupt∈(t1,t0)‖u(⋅,t)‖Wμ,2(Bϱ)s−p≤C(‖u‖Lp(t1,t0;W1,p(Bϱ))+[Diu]λ,0,p;Bϱ×(t1,t0))psupt∈(t1,t0)‖u(⋅,t)‖Wμ,2(Bϱ)s−p.\begin{array}{rcl}\underset{{t}_{1}}{\overset{{t}_{0}}{\displaystyle \int }}\mathop{\displaystyle \int }\limits_{{B}_{r}}{| {D}_{i}u| }^{s}{\rm{d}}x{\rm{d}}t& \le & C\underset{{t}_{1}}{\overset{{t}_{0}}{\displaystyle \int }}{({\Vert u\left(\cdot ,t)\Vert }_{{W}^{1,p}\left({B}_{\varrho })}+{{[}{D}_{i}u\left(\cdot ,t)]}_{\lambda ,p;{B}_{\varrho }})}^{p}{\Vert u\left(\cdot ,t)\Vert }_{{W}^{\mu ,2}\left({B}_{\varrho })}^{s-p}{\rm{d}}t\\ & \le & C\underset{{t}_{1}}{\overset{{t}_{0}}{\displaystyle \int }}{({\Vert u\left(\cdot ,t)\Vert }_{{W}^{1,p}\left({B}_{\varrho })}+{{[}{D}_{i}u\left(\cdot ,t)]}_{\lambda ,p;{B}_{\varrho }})}^{p}{\rm{d}}t\mathop{\sup }\limits_{t\in \left({t}_{1},{t}_{0})}{\Vert u\left(\cdot ,t)\Vert }_{{W}^{\mu ,2}\left({B}_{\varrho })}^{s-p}\\ & \le & C{({\Vert u\Vert }_{{L}^{p}\left({t}_{1},{t}_{0};{W}^{1,p}\left({B}_{\varrho }))}+{{[}{D}_{i}u]}_{\lambda ,0,p;{B}_{\varrho }\times \left({t}_{1},{t}_{0})})}^{p}\mathop{\sup }\limits_{t\in \left({t}_{1},{t}_{0})}{\Vert u\left(\cdot ,t)\Vert }_{{W}^{\mu ,2}\left({B}_{\varrho })}^{s-p}.\end{array}In these calculations, the constant CCdepends on n,μ,λ,p,s,θ,1ϱ−rn,\mu ,\lambda ,p,s,\theta ,\frac{1}{\varrho -r}. Since θ=θ(p,s)\theta =\theta \left(p,s), we ultimately have C=C(n,μ,λ,p,s,1/(ϱ−r))C=C\left(n,\mu ,\lambda ,p,s,1\hspace{0.1em}\text{/}\hspace{0.1em}\left(\varrho -r)). This proves the assertion of the lemma.□We will also need the following embedding results for parabolic Nikolskii spaces, which are defined via finite differences. The first part follows from [3, 7.73], the second part is taken from [12, Proposition 2.19]. Roughly speaking, the lemma asserts that uubelongs to certain fractional Sobolev spaces if certain integral norms of a “fractional difference quotient” are uniformly bounded with respect to hh.Lemma 2.12Let Qϱ(z0)⋐ΩT{Q}_{\varrho }\left({z}_{0})\hspace{0.33em}\Subset \hspace{0.33em}{\Omega }_{T}and θ∈(0,1)\theta \in \left(0,1). (1)Assume that u∈L∞(0,T;L2(Ω,RN))u\in {L}^{\infty }(0,T;\hspace{0.33em}{L}^{2}(\Omega ,{{\mathbb{R}}}^{N}))satisfiessupt∈(t0−ϱ2,t0)∫Bϱ(x0)∣u(x+hei,t)−u(x,t)∣2dx≤M∣h∣2θ\mathop{\sup }\limits_{t\in \left({t}_{0}-{\varrho }^{2},{t}_{0})}\mathop{\int }\limits_{{B}_{\varrho }\left({x}_{0})}{| u\left(x+h{e}_{i},t)-u\left(x,t)| }^{2}{\rm{d}}x\le M{| h| }^{2\theta }for every i∈{1,…,n}i\in \{1,\ldots ,n\}and every h∈Rh\in {\mathbb{R}}with ∣h∣<dist(Bϱ(x0),∂Ω)| h| \lt {\rm{dist}}\left({B}_{\varrho }\left({x}_{0}),\partial \Omega ), where M>0M\gt 0is some constant. Then, for every α∈(0,θ)\alpha \in \left(0,\theta )and O⋐Bϱ(x0){\mathcal{O}}\hspace{0.33em}\Subset \hspace{0.33em}{B}_{\varrho }\left({x}_{0}), there exists a constant C(n,θ,α,dist(O,∂Bϱ),dist(Bϱ,∂Ω))C\left(n,\theta ,\alpha ,{\rm{dist}}\left({\mathcal{O}},\partial {B}_{\varrho }),{\rm{dist}}\left({B}_{\varrho },\partial \Omega ))such thatsupt∈(t0−ϱ2,t0)[u(⋅,t)]α,p;O2=supt∈(t0−ϱ2,t0)∫O∫O∣u(x,t)−u(y,t)∣2∣x−y∣n+2αdxdy≤CM.\mathop{\sup }\limits_{t\in \left({t}_{0}-{\varrho }^{2},{t}_{0})}{{[}u\left(\cdot ,t)]}_{\alpha ,p;\hspace{0.33em}{\mathcal{O}}}^{2}=\mathop{\sup }\limits_{t\in \left({t}_{0}-{\varrho }^{2},{t}_{0})}\mathop{\int }\limits_{{\mathcal{O}}}\mathop{\int }\limits_{{\mathcal{O}}}\frac{{| u\left(x,t)-u(y,t)| }^{2}}{{| x-y| }^{n+2\alpha }}{\rm{d}}x{\rm{d}}y\le CM.(2)Assume that u∈Lp(ΩT,RN)u\in {L}^{p}({\Omega }_{T},{{\mathbb{R}}}^{N})satisfies∫t0−ϱ2t0∫Bϱ(x0)∣u(x+hei,t)−u(x,t)∣pdxdt≤M∣h∣θp\underset{{t}_{0}-{\varrho }^{2}}{\overset{{t}_{0}}{\int }}\mathop{\int }\limits_{{B}_{\varrho }\left({x}_{0})}{| u\left(x+h{e}_{i},t)-u\left(x,t)| }^{p}{\rm{d}}x{\rm{d}}t\le M{| h| }^{\theta p}for every i∈{1,…,n}i\in \{1,\ldots ,n\}and every h∈Rh\in {\mathbb{R}}with ∣h∣<dist(Bϱ(x0),∂Ω)| h| \lt {\rm{dist}}\left({B}_{\varrho }\left({x}_{0}),\partial \Omega ), where M>0M\gt 0is some constant. Then, for every γ∈(0,θ)\gamma \in \left(0,\theta )and O⋐Bϱ(x0){\mathcal{O}}\hspace{0.33em}\Subset \hspace{0.33em}{B}_{\varrho }\left({x}_{0}), there exists a constant C(n,θ,γ,dist(O,∂Bϱ),dist(Bϱ,∂Ω))C\left(n,\theta ,\gamma ,{\rm{dist}}\left({\mathcal{O}},\partial {B}_{\varrho }),{\rm{dist}}\left({B}_{\varrho },\partial \Omega ))such that[u]γ,0,p;O×(t0−ϱ2,t0)p=∫t0−ϱ2t0∫O∫O∣u(x,t)−u(y,t)∣p∣x−y∣n+γpdxdydt≤CM.{\left[u]}_{\gamma ,0,p;\hspace{0.33em}{\mathcal{O}}\times \left({t}_{0}-{\varrho }^{2},{t}_{0})}^{p}=\underset{{t}_{0}-{\varrho }^{2}}{\overset{{t}_{0}}{\int }}\mathop{\int }\limits_{{\mathcal{O}}}\mathop{\int }\limits_{{\mathcal{O}}}\frac{{| u\left(x,t)-u(y,t)| }^{p}}{{| x-y| }^{n+\gamma p}}{\rm{d}}x{\rm{d}}y{\rm{d}}t\le CM.3Caccioppoli inequality for finite differencesIn this section, we prove a Caccioppoli type inequality for finite differences, which will be the starting point for the improvement of integrability. The precise formulation reads as follows:Lemma 3.1Let u be a weak solution of (1.1). There exists a constant C(L,ν,pi)C\left(L,\nu ,{p}_{i})such that for any parabolic cylinder Qϱ(z0)⊂ΩT{Q}_{\varrho }\left({z}_{0})\subset {\Omega }_{T}, any radius r∈(0,ϱ)r\in \left(0,\varrho ), any hhwith ∣h∣<dist(Bϱ(x0),∂Ω)| h| \lt {\rm{dist}}\left({B}_{\varrho }\left({x}_{0}),\partial \Omega ), and any direction s∈{1,…,n}s\in \{1,\ldots ,n\}, the following inequality holds: (3.1)supϑ∈(t0−r2,t0)∫Br(x0)∣τs,hu(⋅,ϑ)∣2dx+∑i=1n∫Qr(z0)∣τs,hDiu∣pidz≤C(ϱ−r)2∑i=1n∫Qϱ(z0)(1+∣Diu∣+∣τs,hDiu∣)pi−2∣τs,hu∣2dz.\begin{array}{l}\mathop{\sup }\limits_{{\vartheta }\in \left({t}_{0}-{r}^{2},{t}_{0})}\mathop{\displaystyle \int }\limits_{{B}_{r}\left({x}_{0})}{| {\tau }_{s,h}u\left(\cdot ,{\vartheta })| }^{2}{\rm{d}}x+\mathop{\displaystyle \sum }\limits_{i=1}^{n}\mathop{\displaystyle \int }\limits_{{Q}_{r}\left({z}_{0})}{| {\tau }_{s,h}{D}_{i}u| }^{{p}_{i}}{\rm{d}}z\\ \hspace{1.0em}\le \frac{C}{{\left(\varrho -r)}^{2}}\mathop{\displaystyle \sum }\limits_{i=1}^{n}\mathop{\displaystyle \int }\limits_{{Q}_{\varrho }\left({z}_{0})}{\left(1+| {D}_{i}u| +| {\tau }_{s,h}{D}_{i}u| )}^{{p}_{i}-2}{| {\tau }_{s,h}u| }^{2}{\rm{d}}z.\end{array}ProofTo simplify the notation, we will w.l.o.g. assume that z0=(x0,t0)=(0,0){z}_{0}=\left({x}_{0},{t}_{0})=\left(0,0)and write Qϱ,Bϱ{Q}_{\varrho },{B}_{\varrho }instead of Qϱ(0),Bϱ(0){Q}_{\varrho }\left(0),{B}_{\varrho }\left(0). Let φ∈C0∞(Qϱ,RN)\varphi \in {C}_{0}^{\infty }({Q}_{\varrho },{{\mathbb{R}}}^{N})and ∣h∣<dist(Bϱ,∂Ω)| h| \lt {\rm{dist}}\left({B}_{\varrho },\partial \Omega ). Then we have τs,−hφ∈C0∞(ΩT,RN){\tau }_{s,-h}\varphi \in {C}_{0}^{\infty }({\Omega }_{T},{{\mathbb{R}}}^{N}). By inserting τs,−hφ{\tau }_{s,-h}\varphi into the weak formulation (1.3) and carrying out an “integration by parts for finite differences,” we obtain (3.2)0=∫ΩTu⋅∂t(τs,−hφ)−⟨Df(Du),D(τs,−hφ)⟩dz=∫Qϱτs,hu⋅∂tφ−⟨τs,hDf(Du),Dφ⟩dz.0=\mathop{\int }\limits_{{\Omega }_{T}}u\cdot {\partial }_{t}\left({\tau }_{s,-h}\varphi )-\langle Df\left(Du),D\left({\tau }_{s,-h}\varphi )\rangle {\rm{d}}z=\mathop{\int }\limits_{{Q}_{\varrho }}{\tau }_{s,h}u\cdot {\partial }_{t}\varphi -\langle {\tau }_{s,h}Df\left(Du),D\varphi \rangle {\rm{d}}z.Now we want to insert suitable test functions into (3.2). With respect to space, we choose a cutoff function η∈C0∞(Bϱ)\eta \in {C}_{0}^{\infty }\left({B}_{\varrho })such that 0≤η≤10\le \eta \le 1, η=1\eta =1in Br{B}_{r}and (3.3)∣Dη∣≤2ϱ−r.| D\eta | \le \frac{2}{\varrho -r}.With respect to time, we define a cut-off function ζε∈W01,∞(−ϱ2,0){\zeta }_{\varepsilon }\in {W}_{0}^{1,\infty }(-{\varrho }^{2},0)via (3.4)ζε(t)=1ϱ2−r2(t+ϱ2),on(−ϱ2,−r2),1,on(−r2,ϑ),1ε(ϑ+ε−t),on(ϑ,ϑ+ε),0,on(ϑ+ε,0),{\zeta }_{\varepsilon }\left(t)=\left\{\begin{array}{ll}\frac{1}{{\varrho }^{2}-{r}^{2}}(t+{\varrho }^{2}),\hspace{1.0em}& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}(-{\varrho }^{2},-{r}^{2}),\\ 1,\hspace{1.0em}& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\left(-{r}^{2},{\vartheta }),\\ \frac{1}{\varepsilon }({\vartheta }+\varepsilon -t),\hspace{1.0em}& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\left({\vartheta },{\vartheta }+\varepsilon ),\\ 0,\hspace{1.0em}& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\left({\vartheta }+\varepsilon ,0),\end{array}\right.for some ϑ∈(−r2,0){\vartheta }\in \left(-{r}^{2},0)and ε∈(0,∣ϑ∣)\varepsilon \in \left(0,| {\vartheta }| ). As a test function in the modified weak formulation (3.2), we choose φε(x,t)≔τs,hu(x,t)η2(x)ζε(t).{\varphi }_{\varepsilon }\left(x,t):= {\tau }_{s,h}u\left(x,t){\eta }^{2}\left(x){\zeta }_{\varepsilon }\left(t).Note that this is actually not an admissible test function since it is not smooth and does not even possess a weak derivative with respect to time. The following formal computations, however, can be made rigorous with a standard smoothing procedure with respect to time, as, for instance, via Steklov averages. By inserting φε{\varphi }_{\varepsilon }into (3.2), we obtain −∫Qϱτs,hu⋅∂t(τs,huη2ζε)dz+∫Qϱ⟨τs,hDf(Du),τs,hDu⟩η2ζεdz=−2∫Qϱ⟨τs,hDf(Du),τs,hu⊗Dη⟩ηζεdz.-\mathop{\int }\limits_{{Q}_{\varrho }}{\tau }_{s,h}u\cdot {\partial }_{t}({\tau }_{s,h}u\hspace{0.33em}{\eta }^{2}{\zeta }_{\varepsilon }){\rm{d}}z+\mathop{\int }\limits_{{Q}_{\varrho }}\langle {\tau }_{s,h}Df\left(Du),{\tau }_{s,h}Du\rangle {\eta }^{2}{\zeta }_{\varepsilon }{\rm{d}}z=-2\mathop{\int }\limits_{{Q}_{\varrho }}\langle {\tau }_{s,h}Df\left(Du),{\tau }_{s,h}u\otimes D\eta \rangle \eta {\zeta }_{\varepsilon }{\rm{d}}z.For the first term on the left-hand side, we calculate −∫Qϱτs,hu⋅∂t(τs,huη2ζε)dz=−∫Qϱ∣τs,hu∣2∂tζεη2dz−∫Qϱτs,hu⋅∂t(τs,hu)η2ζεdz=−∫Qϱ∣τs,hu∣2∂tζεη2dz−12∫Qϱ∂t(∣τs,hu∣2)η2ζεdz=−12∫Qϱ∣τs,hu∣2∂tζεη2dz=−12(ϱ2−r2)∫−ϱ2−r2∫Bϱ∣τs,hu∣2η2dxdt+12ε∫ϑϑ+ε∫Bϱ∣τs,hu∣2η2dxdt.\begin{array}{rcl}-\mathop{\displaystyle \int }\limits_{{Q}_{\varrho }}{\tau }_{s,h}u\cdot {\partial }_{t}({\tau }_{s,h}u{\eta }^{2}{\zeta }_{\varepsilon }){\rm{d}}z& =& -\mathop{\displaystyle \int }\limits_{{Q}_{\varrho }}{| {\tau }_{s,h}u| }^{2}{\partial }_{t}{\zeta }_{\varepsilon }{\eta }^{2}{\rm{d}}z-\mathop{\displaystyle \int }\limits_{{Q}_{\varrho }}{\tau }_{s,h}u\cdot {\partial }_{t}\left({\tau }_{s,h}u){\eta }^{2}{\zeta }_{\varepsilon }{\rm{d}}z\\ & =& -\mathop{\displaystyle \int }\limits_{{Q}_{\varrho }}{| {\tau }_{s,h}u| }^{2}{\partial }_{t}{\zeta }_{\varepsilon }{\eta }^{2}{\rm{d}}z-\frac{1}{2}\mathop{\displaystyle \int }\limits_{{Q}_{\varrho }}{\partial }_{t}({| {\tau }_{s,h}u| }^{2}){\eta }^{2}{\zeta }_{\varepsilon }{\rm{d}}z\\ & =& -\frac{1}{2}\mathop{\displaystyle \int }\limits_{{Q}_{\varrho }}{| {\tau }_{s,h}u| }^{2}{\partial }_{t}{\zeta }_{\varepsilon }{\eta }^{2}{\rm{d}}z\\ & =& -\frac{1}{2({\varrho }^{2}-{r}^{2})}\underset{-{\varrho }^{2}}{\overset{-{r}^{2}}{\displaystyle \int }}\mathop{\displaystyle \int }\limits_{{B}_{\varrho }}{| {\tau }_{s,h}u| }^{2}{\eta }^{2}{\rm{d}}x{\rm{d}}t+\frac{1}{2\varepsilon }\underset{{\vartheta }}{\overset{{\vartheta }+\varepsilon }{\displaystyle \int }}\mathop{\displaystyle \int }\limits_{{B}_{\varrho }}{| {\tau }_{s,h}u| }^{2}{\eta }^{2}{\rm{d}}x{\rm{d}}t.\end{array}By using the estimate 12(ϱ2−r2)≤1(ϱ−r)2\frac{1}{2\left({\varrho }^{2}-{r}^{2})}\le \frac{1}{{\left(\varrho -r)}^{2}}, we obtain (3.5)I+II≔12ε∫ϑϑ+ε∫Bϱ∣τs,hu∣2η2dxdt+∫Qϱ⟨τs,hDf(Du),τs,hDu⟩η2ζεdz≤1(ϱ−r)2∫−ϱ2−r2∫Bϱ∣τs,hu∣2η2dxdt−2∫Qϱ⟨τs,hDf(Du),τs,hu⊗Dη⟩ηζεdz≕III+IV,\begin{array}{rcl}I+II& := & \frac{1}{2\varepsilon }\underset{{\vartheta }}{\overset{{\vartheta }+\varepsilon }{\displaystyle \int }}\mathop{\displaystyle \int }\limits_{{B}_{\varrho }}{| {\tau }_{s,h}u| }^{2}{\eta }^{2}{\rm{d}}x{\rm{d}}t+\mathop{\displaystyle \int }\limits_{{Q}_{\varrho }}\langle {\tau }_{s,h}Df\left(Du),{\tau }_{s,h}Du\rangle {\eta }^{2}{\zeta }_{\varepsilon }{\rm{d}}z\\ & \le & \frac{1}{{\left(\varrho -r)}^{2}}\underset{-{\varrho }^{2}}{\overset{-{r}^{2}}{\displaystyle \int }}\mathop{\displaystyle \int }\limits_{{B}_{\varrho }}{| {\tau }_{s,h}u| }^{2}{\eta }^{2}{\rm{d}}x{\rm{d}}t-2\mathop{\displaystyle \int }\limits_{{Q}_{\varrho }}\langle {\tau }_{s,h}Df\left(Du),{\tau }_{s,h}u\displaystyle \otimes D\eta \rangle \eta {\zeta }_{\varepsilon }{\rm{d}}z\hspace{0.33em}=: \hspace{0.33em}III+IV,\end{array}with the obvious meaning of II–IVIV. Due to the properties of the cutoff function η\eta , we have I≥12ε∫ϑϑ+ε∫Br∣τs,hu∣2dxdtI\ge \frac{1}{2\varepsilon }\underset{{\vartheta }}{\overset{{\vartheta }+\varepsilon }{\int }}\mathop{\int }\limits_{{B}_{r}}{| {\tau }_{s,h}u| }^{2}{\rm{d}}x{\rm{d}}tand ∣III∣≤1(ϱ−r)2∫Qϱ∣τs,hu∣2dz.| III| \le \frac{1}{{\left(\varrho -r)}^{2}}\mathop{\int }\limits_{{Q}_{\varrho }}{| {\tau }_{s,h}u| }^{2}{\rm{d}}z.The second term can be rewritten in the following way: II=∫Qϱ∫01ddα[Df(Du+ατs,hDu)],τs,hDuη2ζεdαdz=∫Qϱ∫01⟨D2f(Du+ατs,hDu)τs,hDu,τs,hDu⟩η2ζεdαdz.\begin{array}{rcl}II& =& \mathop{\displaystyle \int }\limits_{{Q}_{\varrho }}\underset{0}{\overset{1}{\displaystyle \int }}\left\langle \frac{{\rm{d}}}{{\rm{d}}\alpha }{[}Df\left(Du+\alpha {\tau }_{s,h}Du)],{\tau }_{s,h}Du\right\rangle {\eta }^{2}{\zeta }_{\varepsilon }{\rm{d}}\alpha {\rm{d}}z\\ & =& \mathop{\displaystyle \int }\limits_{{Q}_{\varrho }}\underset{0}{\overset{1}{\displaystyle \int }}\langle {D}^{2}f\left(Du+\alpha {\tau }_{s,h}Du){\tau }_{s,h}Du,{\tau }_{s,h}Du\rangle {\eta }^{2}{\zeta }_{\varepsilon }{\rm{d}}\alpha {\rm{d}}z.\end{array}Similarly, the fourth term can be rewritten in the following way: IV=∫Qϱ∫01⟨D2f(Du+ατs,hDu)τs,hDu,τs,hu⊗Dη⟩ηζεdαdz.IV=\mathop{\int }\limits_{{Q}_{\varrho }}\underset{0}{\overset{1}{\int }}\langle {D}^{2}f\left(Du+\alpha {\tau }_{s,h}Du){\tau }_{s,h}Du,{\tau }_{s,h}u\otimes D\eta \rangle \eta {\zeta }_{\varepsilon }{\rm{d}}\alpha {\rm{d}}z.Now we use the Cauchy-Schwarz inequality for the symmetric bilinear form A(σ,σ˜)≔⟨D2f(Du+ατs,hDu)σ,σ˜⟩{\mathcal{A}}\left(\sigma ,\tilde{\sigma }):= \langle {D}^{2}f\left(Du+\alpha {\tau }_{s,h}Du)\sigma ,\tilde{\sigma }\rangle and Young’s inequality to obtain ∣IV∣≤∫Qϱ∫01A(τs,hDu,τs,hDu)A(τs,hu⊗Dη,τs,hu⊗Dη)ηζεdαdz≤12II+12∫Qϱ∫01⟨D2f(Du+ατs,hDu)τs,hu⊗Dη,τs,hu⊗Dη⟩ζεdαdz≤12II+2L(ϱ−r)2∫Qϱ∫011+∑i=1n∣Diu+ατs,hDiu∣pi−2∣τs,hu∣2dαdz,\begin{array}{rcl}| IV| & \le & \mathop{\displaystyle \int }\limits_{{Q}_{\varrho }}\underset{0}{\overset{1}{\displaystyle \int }}\sqrt{{\mathcal{A}}\left({\tau }_{s,h}Du,{\tau }_{s,h}Du)}\sqrt{{\mathcal{A}}\left({\tau }_{s,h}u\displaystyle \otimes D\eta ,{\tau }_{s,h}u\displaystyle \otimes D\eta )}\eta {\zeta }_{\varepsilon }{\rm{d}}\alpha {\rm{d}}z\\ & \le & \frac{1}{2}II+\frac{1}{2}\mathop{\displaystyle \int }\limits_{{Q}_{\varrho }}\underset{0}{\overset{1}{\displaystyle \int }}\langle {D}^{2}f\left(Du+\alpha {\tau }_{s,h}Du){\tau }_{s,h}u\displaystyle \otimes D\eta ,{\tau }_{s,h}u\displaystyle \otimes D\eta \rangle {\zeta }_{\varepsilon }{\rm{d}}\alpha {\rm{d}}z\\ & \le & \frac{1}{2}II+\frac{2L}{{\left(\varrho -r)}^{2}}\mathop{\displaystyle \int }\limits_{{Q}_{\varrho }}\underset{0}{\overset{1}{\displaystyle \int }}\left(1+\mathop{\displaystyle \sum }\limits_{i=1}^{n}{| {D}_{i}u+\alpha {\tau }_{s,h}{D}_{i}u| }^{{p}_{i}-2}\right){| {\tau }_{s,h}u| }^{2}{\rm{d}}\alpha {\rm{d}}z,\end{array}where we used the estimate (3.3) for ∣Dη∣| D\eta | , the growth condition (1.2)2{}_{2}, and the fact that ζε≤1{\zeta }_{\varepsilon }\le 1. By inserting the estimates for the terms II–IVIVinto (3.5) and absorbing the term 12II\frac{1}{2}IIon the left-hand side, we obtain 12ε∫ϑϑ+ε∫Br∣τs,hu∣2dxdt+12∫Qϱ∫01⟨D2f(Du+ατs,hDu)τs,hDu,τs,hDu⟩η2ζεdαdz≤C(L)(ϱ−r)2∫Qϱ∫011+∑i=1n∣Diu+ατs,hDiu∣pi−2∣τs,hu∣2dαdz≤C(L)(ϱ−r)2∑i=1n∫Qϱ∫01(1+∣Diu+ατs,hDiu∣pi−2)dα∣τs,hu∣2dz.\begin{array}{l}\frac{1}{2\varepsilon }\underset{{\vartheta }}{\overset{{\vartheta }+\varepsilon }{\displaystyle \int }}\mathop{\displaystyle \int }\limits_{{B}_{r}}{| {\tau }_{s,h}u| }^{2}{\rm{d}}x{\rm{d}}t+\frac{1}{2}\mathop{\displaystyle \int }\limits_{{Q}_{\varrho }}\underset{0}{\overset{1}{\displaystyle \int }}\langle {D}^{2}f\left(Du+\alpha {\tau }_{s,h}Du){\tau }_{s,h}Du,{\tau }_{s,h}Du\rangle {\eta }^{2}{\zeta }_{\varepsilon }{\rm{d}}\alpha {\rm{d}}z\\ \hspace{1.0em}\le \frac{C\left(L)}{{\left(\varrho -r)}^{2}}\mathop{\displaystyle \int }\limits_{{Q}_{\varrho }}\underset{0}{\overset{1}{\displaystyle \int }}\left(1+\mathop{\displaystyle \sum }\limits_{i=1}^{n}{| {D}_{i}u+\alpha {\tau }_{s,h}{D}_{i}u| }^{{p}_{i}-2}\right){| {\tau }_{s,h}u| }^{2}{\rm{d}}\alpha {\rm{d}}z\\ \hspace{1.0em}\le \frac{C\left(L)}{{\left(\varrho -r)}^{2}}\mathop{\displaystyle \sum }\limits_{i=1}^{n}\mathop{\displaystyle \int }\limits_{{Q}_{\varrho }}\underset{0}{\overset{1}{\displaystyle \int }}(1+{| {D}_{i}u+\alpha {\tau }_{s,h}{D}_{i}u| }^{{p}_{i}-2}){\rm{d}}\alpha {| {\tau }_{s,h}u| }^{2}{\rm{d}}z.\end{array}Now we use the ellipticity condition (1.2)3{}_{3}and apply Lemma 2.2 (with A=Diu(x,t)A={D}_{i}u\left(x,t), B=Diu(x+hes,t)B={D}_{i}u\left(x+h{e}_{s},t), σ=pi−22\sigma =\frac{{p}_{i}-2}{2}, and μ=0\mu =0) to estimate the second term on the left-hand side from below: ∫Qϱ∫01⟨D2f(Du+ατs,hDu)τs,hDu,τs,hDu⟩η2ζεdαdz≥ν∫Qϱ∫01∑i=1n∣Diu+ατs,hDiu∣pi−2∣τs,hDiu∣2η2ζεdαdz≥νC(pi)∑i=1n∫Qϱ(∣Diu(x,t)∣2+∣Diu(x+hes,t)∣2)pi−22∣τs,hDiu∣2η2ζεdz≥νC(pi)∑i=1n∫Qϱ∣τs,hDiu∣piη2ζεdz≥νC(pi)∑i=1n∫Br×(−r2,ϑ)∣τs,hDiu∣pidz.\begin{array}{l}\mathop{\displaystyle \int }\limits_{{Q}_{\varrho }}\underset{0}{\overset{1}{\displaystyle \int }}\langle {D}^{2}f\left(Du+\alpha {\tau }_{s,h}Du){\tau }_{s,h}Du,{\tau }_{s,h}Du\rangle {\eta }^{2}{\zeta }_{\varepsilon }{\rm{d}}\alpha {\rm{d}}z\\ \hspace{1.0em}\ge \nu \mathop{\displaystyle \int }\limits_{{Q}_{\varrho }}\underset{0}{\overset{1}{\displaystyle \int }}\left(\mathop{\displaystyle \sum }\limits_{i=1}^{n}{| {D}_{i}u+\alpha {\tau }_{s,h}{D}_{i}u| }^{{p}_{i}-2}{| {\tau }_{s,h}{D}_{i}u| }^{2}\right){\eta }^{2}{\zeta }_{\varepsilon }{\rm{d}}\alpha {\rm{d}}z\\ \hspace{1.0em}\ge \frac{\nu }{C\left({p}_{i})}\mathop{\displaystyle \sum }\limits_{i=1}^{n}\mathop{\displaystyle \int }\limits_{{Q}_{\varrho }}{({| {D}_{i}u\left(x,t)| }^{2}+{| {D}_{i}u\left(x+h{e}_{s},t)| }^{2})}^{\tfrac{{p}_{i}-2}{2}}{| {\tau }_{s,h}{D}_{i}u| }^{2}{\eta }^{2}{\zeta }_{\varepsilon }{\rm{d}}z\\ \hspace{1.0em}\ge \frac{\nu }{C\left({p}_{i})}\mathop{\displaystyle \sum }\limits_{i=1}^{n}\mathop{\displaystyle \int }\limits_{{Q}_{\varrho }}{| {\tau }_{s,h}{D}_{i}u| }^{{p}_{i}}{\eta }^{2}{\zeta }_{\varepsilon }{\rm{d}}z\\ \hspace{1.0em}\ge \frac{\nu }{C\left({p}_{i})}\mathop{\displaystyle \sum }\limits_{i=1}^{n}\mathop{\displaystyle \int }\limits_{{B}_{r}\times \left(-{r}^{2},{\vartheta })}{| {\tau }_{s,h}{D}_{i}u| }^{{p}_{i}}{\rm{d}}z.\end{array}In the last step, we also used the definition of η\eta and the definition of ζε{\zeta }_{\varepsilon }from (3.4). On the other hand, due to Lemma 2.2 (applied with μ=1\mu =1and A,B,σA,B,\sigma as mentioned earlier) we also have ∫01(1+∣Diu+ατs,hDiu∣pi−2)dα≤2∫01(1+∣Diu+ατs,hDiu∣2)pi−22dα≤C(pi)(1+∣Diu(x,t)∣2+∣Diu(x+hes,t)∣2)pi−22≤C(pi)(1+∣Diu(x,t)∣2+∣Diu(x+hes,t)−Diu(x,t)∣2)pi−22≤C(pi)(1+∣Diu∣+∣τs,hDiu∣)pi−2.\begin{array}{rcl}\underset{0}{\overset{1}{\displaystyle \int }}(1+{| {D}_{i}u+\alpha {\tau }_{s,h}{D}_{i}u| }^{{p}_{i}-2}){\rm{d}}\alpha & \le & 2\underset{0}{\overset{1}{\displaystyle \int }}{(1+{| {D}_{i}u+\alpha {\tau }_{s,h}{D}_{i}u| }^{2})}^{\tfrac{{p}_{i}-2}{2}}{\rm{d}}\alpha \\ & \le & C\left({p}_{i}){(1+{| {D}_{i}u\left(x,t)| }^{2}+{| {D}_{i}u\left(x+h{e}_{s},t)| }^{2})}^{\tfrac{{p}_{i}-2}{2}}\\ & \le & C\left({p}_{i}){(1+{| {D}_{i}u\left(x,t)| }^{2}+{| {D}_{i}u\left(x+h{e}_{s},t)-{D}_{i}u\left(x,t)| }^{2})}^{\tfrac{{p}_{i}-2}{2}}\\ & \le & C\left({p}_{i}){(1+| {D}_{i}u| +| {\tau }_{s,h}{D}_{i}u| )}^{{p}_{i}-2}.\end{array}By putting the previous estimates together, we obtain 1ε∫ϑϑ+ε∫Br∣τs,hu∣2dxdt+∑i=1n∫Br×(−r2,ϑ)∣τs,hDiu∣pidz≤C(L,ν,pi)(ϱ−r)2∑i=1n∫Qϱ(1+∣Diu∣+∣τs,hDiu∣)pi−2∣τs,hu∣2dz.\frac{1}{\varepsilon }\underset{{\vartheta }}{\overset{{\vartheta }+\varepsilon }{\int }}\mathop{\int }\limits_{{B}_{r}}{| {\tau }_{s,h}u| }^{2}{\rm{d}}x{\rm{d}}t+\mathop{\sum }\limits_{i=1}^{n}\mathop{\int }\limits_{{B}_{r}\times \left(-{r}^{2},{\vartheta })}{| {\tau }_{s,h}{D}_{i}u| }^{{p}_{i}}{\rm{d}}z\le \frac{C\left(L,\nu ,{p}_{i})}{{\left(\varrho -r)}^{2}}\mathop{\sum }\limits_{i=1}^{n}\mathop{\int }\limits_{{Q}_{\varrho }}{\left(1+| {D}_{i}u| +| {\tau }_{s,h}{D}_{i}u| )}^{{p}_{i}-2}{| {\tau }_{s,h}u| }^{2}{\rm{d}}z.The first term on the left-hand side converges to ∫Br∣τs,hu(⋅,ϑ)∣2dx{\int }_{{B}_{r}}{| {\tau }_{s,h}u\left(\cdot ,{\vartheta })| }^{2}{\rm{d}}xas ε→0\varepsilon \to 0. Hence, going to the supremum with respect to ϑ{\vartheta }in the first term on the left-hand side, we obtain that supϑ∈(−r2,0)∫Br∣τs,hu(⋅,ϑ)∣2dx≤C(L,ν,pi)(ϱ−r)2∑i=1n∫Qϱ(1+∣Diu∣+∣τs,hDiu∣)pi−2∣τs,hu∣2dz.\mathop{\sup }\limits_{{\vartheta }\in \left(-{r}^{2},0)}\mathop{\int }\limits_{{B}_{r}}{| {\tau }_{s,h}u\left(\cdot ,{\vartheta })| }^{2}{\rm{d}}x\le \frac{C\left(L,\nu ,{p}_{i})}{{\left(\varrho -r)}^{2}}\mathop{\sum }\limits_{i=1}^{n}\mathop{\int }\limits_{{Q}_{\varrho }}{\left(1+| {D}_{i}u| +| {\tau }_{s,h}{D}_{i}u| )}^{{p}_{i}-2}{| {\tau }_{s,h}u| }^{2}{\rm{d}}z.In the second term on the left-hand side, we let ϑ→0{\vartheta }\to 0to obtain ∑i=1n∫Br×(−r2,0)∣τs,hDiu∣pidz≤C(L,ν,pi)(ϱ−r)2∑i=1n∫Qϱ(1+∣Diu∣+∣τs,hDiu∣)pi−2∣τs,hu∣2dz.\mathop{\sum }\limits_{i=1}^{n}\mathop{\int }\limits_{{B}_{r}\times \left(-{r}^{2},0)}{| {\tau }_{s,h}{D}_{i}u| }^{{p}_{i}}{\rm{d}}z\le \frac{C\left(L,\nu ,{p}_{i})}{{\left(\varrho -r)}^{2}}\mathop{\sum }\limits_{i=1}^{n}\mathop{\int }\limits_{{Q}_{\varrho }}{\left(1+| {D}_{i}u| +| {\tau }_{s,h}{D}_{i}u| )}^{{p}_{i}-2}{| {\tau }_{s,h}u| }^{2}{\rm{d}}z.By adding the two previous inequalities, we obtain the desired inequality □supϑ∈(−r2,0)∫Br∣τs,hu(⋅,ϑ)∣2dx+∑i=1n∫Qr∣τs,hDiu∣pidz≤C(L,ν,pi)(ϱ−r)2∑i=1n∫Qϱ(1+∣Diu∣+∣τs,hDiu∣)pi−2∣τs,hu∣2dz.\mathop{\sup }\limits_{{\vartheta }\in \left(-{r}^{2},0)}\mathop{\int }\limits_{{B}_{r}}{| {\tau }_{s,h}u\left(\cdot ,{\vartheta })| }^{2}{\rm{d}}x+\mathop{\sum }\limits_{i=1}^{n}\mathop{\int }\limits_{{Q}_{r}}{| {\tau }_{s,h}{D}_{i}u| }^{{p}_{i}}{\rm{d}}z\le \frac{C\left(L,\nu ,{p}_{i})}{{\left(\varrho -r)}^{2}}\mathop{\sum }\limits_{i=1}^{n}\mathop{\int }\limits_{{Q}_{\varrho }}{\left(1+| {D}_{i}u| +| {\tau }_{s,h}{D}_{i}u| )}^{{p}_{i}-2}{| {\tau }_{s,h}u| }^{2}{\rm{d}}z.4Proof of the main theoremWe have now acquired all the necessary tools to prove Theorem 1.4. The main idea is to derive a uniform bound (with respect to hh) for the right-hand side of the Caccioppoli inequality (3.1). Lemma 2.12 then asserts that uubelongs to certain fractional parabolic Sobolev spaces. Subsequently, we can use Lemma 2.11 to improve the integrability, of a fixed partial derivative Diu{D}_{i}u. Finally, we need to perform an iteration procedure to obtain the full higher integrability, i.e., Du∈Llocpn+εDu\in {L}_{{\rm{loc}}}^{{p}_{n}+\varepsilon }.Proof of Theorem 1.4Let z0∈ΩT{z}_{0}\in {\Omega }_{T}, let ϱ∈(0,1)\varrho \in \left(0,1)be some radius such that Q2ϱ(z0)⊂ΩT{Q}_{2\varrho }\left({z}_{0})\subset {\Omega }_{T}, and let ϱ2≤ϱ1<ϱ2≤ϱ\frac{\varrho }{2}\le {\varrho }_{1}\lt {\varrho }_{2}\le \varrho . Furthermore, let h∈(−ϱ,ϱ)h\in \left(-\varrho ,\varrho )and let s∈{1,…,n}s\in \{1,\ldots ,n\}be arbitrary but fixed. In the following, we suppress the center of the cylinder in our notation by writing, e.g., Qϱ{Q}_{\varrho }instead of Qϱ(z0){Q}_{\varrho }\left({z}_{0}). Since Q2ϱ⊂ΩT{Q}_{2\varrho }\subset {\Omega }_{T}and ∣h∣<ϱ| h| \lt \varrho , we can apply Lemma 3.1 with ϱ2+ϱ12,ϱ1\left(\frac{{\varrho }_{2}+{\varrho }_{1}}{2},{\varrho }_{1}\right)instead of (ϱ,r)\left(\varrho ,r), which yields the following estimate: (4.1)supt∈(−ϱ12,0)∫Bϱ1∣τs,hu(⋅,t)∣2dx+∑i=1n∫Qϱ1∣τs,hDiu∣pidz≤C(L,ν,pi)(ϱ2−ϱ1)2∑i=1n∫Qϱ2+ϱ12(1+∣Diu∣+∣τs,hDiu∣)pi−2∣τs,hu∣2dz.\mathop{\sup }\limits_{t\in \left(-{\varrho }_{1}^{2},0)}\mathop{\int }\limits_{{B}_{{\varrho }_{1}}}{| {\tau }_{s,h}u\left(\cdot ,t)| }^{2}{\rm{d}}x+\mathop{\sum }\limits_{i=1}^{n}\mathop{\int }\limits_{{Q}_{{\varrho }_{1}}}{| {\tau }_{s,h}{D}_{i}u| }^{{p}_{i}}{\rm{d}}z\le \frac{C\left(L,\nu ,{p}_{i})}{{\left({\varrho }_{2}-{\varrho }_{1})}^{2}}\mathop{\sum }\limits_{i=1}^{n}\mathop{\int }\limits_{{Q}_{\tfrac{{\varrho }_{2}+{\varrho }_{1}}{2}}}{\left(1+| {D}_{i}u| +| {\tau }_{s,h}{D}_{i}u| )}^{{p}_{i}-2}{| {\tau }_{s,h}u| }^{2}{\rm{d}}z.Step 1: Let us assume that Du∈Llocσ(ΩT,RN×n)for someσ∈[p1,pn),Du\in {L}_{\hspace{0.1em}\text{loc}\hspace{0.1em}}^{\sigma }({\Omega }_{T},{{\mathbb{R}}}^{N\times n})\hspace{1.0em}\hspace{0.1em}\text{for some}\hspace{0.1em}\hspace{0.33em}\sigma \in \left[{p}_{1},{p}_{n}),and note that this assumption is satisfied for the choice σ=p1\sigma ={p}_{1}. Now we want to show that the right-hand side of (4.1) can be bounded in terms of some power of ∣h∣| h| , i.e., 1(ϱ2−ϱ1)2∑i=1n∫Qϱ2+ϱ12(1+∣Diu∣+∣τs,hDiu∣)pi−2∣τs,hu∣2dz︸≕RHSi≤C∣h∣2γ\frac{1}{{\left({\varrho }_{2}-{\varrho }_{1})}^{2}}\mathop{\sum }\limits_{i=1}^{n}\mathop{\underbrace{\mathop{\int }\limits_{{Q}_{\tfrac{{\varrho }_{2}+{\varrho }_{1}}{2}}}{\left(1+| {D}_{i}u| +| {\tau }_{s,h}{D}_{i}u| )}^{{p}_{i}-2}{| {\tau }_{s,h}u| }^{2}{\rm{d}}z}}\limits_{=: {\text{RHS}}_{i}}\le C{| h| }^{2\gamma }for some γ∈(0,1)\gamma \in \left(0,1), with a constant CCthat does not depend on hh. We distinguish between two cases: In the case pi≤σ{p}_{i}\le \sigma , we can use Hölder’s inequality to estimate (4.2)RHSi≤∫Qϱ2+ϱ12(1+∣Diu∣+∣τs,hDiu∣)pidzpi−2pi∫Qϱ2+ϱ12∣τs,hu∣pidz2pi≤C(pi)∫Q2ϱ(1+∣Diu∣)pidzpi−2pi∫Q2ϱ∣Dsu∣pidz2pi︸<∞∣h∣2≤C∣h∣2,\begin{array}{rcl}{\text{RHS}}_{i}& \le & {\left(\mathop{\displaystyle \int }\limits_{{Q}_{\tfrac{{\varrho }_{2}+{\varrho }_{1}}{2}}}{\left(1+| {D}_{i}u| +| {\tau }_{s,h}{D}_{i}u| )}^{{p}_{i}}{\rm{d}}z\right)}^{\tfrac{{p}_{i}-2}{{p}_{i}}}{\left(\mathop{\displaystyle \int }\limits_{{Q}_{\tfrac{{\varrho }_{2}+{\varrho }_{1}}{2}}}{| {\tau }_{s,h}u| }^{{p}_{i}}{\rm{d}}z\right)}^{\tfrac{2}{{p}_{i}}}\\ & \le & C\left({p}_{i})\mathop{\underbrace{{\left(\mathop{\displaystyle \int }\limits_{{Q}_{2\varrho }}{\left(1+| {D}_{i}u| )}^{{p}_{i}}{\rm{d}}z\right)}^{\tfrac{{p}_{i}-2}{{p}_{i}}}{\left(\mathop{\displaystyle \int }\limits_{{Q}_{2\varrho }}{| {D}_{s}u| }^{{p}_{i}}{\rm{d}}z\right)}^{\tfrac{2}{{p}_{i}}}}}\limits_{\lt \infty }{| h| }^{2}\\ & \le & C{| h| }^{2},\end{array}where we used (2.1) and (2.2) in the penultimate step. In the last step, we used the fact that Dsu∈Lpi(Q2ϱ){D}_{s}u\in {L}^{{p}_{i}}\left({Q}_{2\varrho })since pi≤σ{p}_{i}\le \sigma and Du∈Llocσ(ΩT)Du\in {L}_{{\rm{loc}}}^{\sigma }\left({\Omega }_{T}). In these calculations, the constant CCdepends on pi{p}_{i}, ‖Diu‖Lpi(ΩT){\Vert {D}_{i}u\Vert }_{{L}^{{p}_{i}}\left({\Omega }_{T})}and ‖Dsu‖Lpi(Q2ϱ){\Vert {D}_{s}u\Vert }_{{L}^{{p}_{i}}\left({Q}_{2\varrho })}, but not on hh. Note that the estimate above also holds in the limit case pi=2{p}_{i}=2.Let us now consider the case pi>σ{p}_{i}\gt \sigma . This case is more difficult to deal with, since we do not know a priori that Dsu∈Lpi(Q2ϱ){D}_{s}u\in {L}^{{p}_{i}}\left({Q}_{2\varrho }). Let (4.3)ai∈0,σp1pi{a}_{i}\in \left(0,\frac{\sigma {p}_{1}}{{p}_{i}}\right)be a parameter, that will be fixed later. We define the exponents q1=pipi−2,q2=σp12ai,q3=σp1pi2(σp1−aipi).{q}_{1}=\frac{{p}_{i}}{{p}_{i}-2},\hspace{1.0em}{q}_{2}=\frac{\sigma {p}_{1}}{2{a}_{i}},\hspace{1.0em}{q}_{3}=\frac{\sigma {p}_{1}{p}_{i}}{2\left(\sigma {p}_{1}-{a}_{i}{p}_{i})}.Note that q1,q2,q3>1{q}_{1},{q}_{2},{q}_{3}\gt 1and 1q1+1q2+1q3=1\frac{1}{{q}_{1}}+\frac{1}{{q}_{2}}+\frac{1}{{q}_{3}}=1. We apply Hölder’s inequality with these exponents and use (2.1) and (2.2) to obtain RHSi=∫Qϱ2+ϱ12(1+∣Diu∣+∣τs,hDiu∣)pi−2∣τs,hu∣2aip1∣τs,hu∣2(p1−ai)p1dz≤∫Qϱ2+ϱ12(1+∣Diu∣+∣τs,hDiu∣)pidzpi−2pi∫Qϱ2+ϱ12∣τs,hu∣σdz2aiσp1∫Qϱ2+ϱ12∣τs,hu∣2q3(p1−ai)p1dz1q3≤C(pi)∫Q2ϱ(1+∣Diu∣)pidzpi−2pi∫Q2ϱ∣Dsu∣σdz2aiσp1∣h∣2aip1∫Qϱ2+ϱ12∣τs,hu∣2q3(p1−ai)p1dz1q3≤C∣h∣2aip1∫Qϱ2+ϱ12∣τs,hu∣2q3(p1−ai)p1dz1q3,\begin{array}{rcl}{\text{RHS}}_{i}& =& \mathop{\displaystyle \int }\limits_{{Q}_{\tfrac{{\varrho }_{2}+{\varrho }_{1}}{2}}}{\left(1+| {D}_{i}u| +| {\tau }_{s,h}{D}_{i}u| )}^{{p}_{i}-2}{| {\tau }_{s,h}u| }^{\tfrac{2{a}_{i}}{{p}_{1}}}{| {\tau }_{s,h}u| }^{\tfrac{2\left({p}_{1}-{a}_{i})}{{p}_{1}}}{\rm{d}}z\\ & \le & {\left(\mathop{\displaystyle \int }\limits_{{Q}_{\tfrac{{\varrho }_{2}+{\varrho }_{1}}{2}}}{\left(1+| {D}_{i}u| +| {\tau }_{s,h}{D}_{i}u| )}^{{p}_{i}}{\rm{d}}z\right)}^{\tfrac{{p}_{i}-2}{{p}_{i}}}{\left(\mathop{\displaystyle \int }\limits_{{Q}_{\tfrac{{\varrho }_{2}+{\varrho }_{1}}{2}}}{| {\tau }_{s,h}u| }^{\sigma }{\rm{d}}z\right)}^{\tfrac{2{a}_{i}}{\sigma {p}_{1}}}{\left(\mathop{\displaystyle \int }\limits_{{Q}_{\tfrac{{\varrho }_{2}+{\varrho }_{1}}{2}}}{| {\tau }_{s,h}u| }^{\frac{2{q}_{3}\left({p}_{1}-{a}_{i})}{{p}_{1}}}{\rm{d}}z\right)}^{\tfrac{1}{{q}_{3}}}\\ & \le & C\left({p}_{i}){\left(\mathop{\displaystyle \int }\limits_{{Q}_{2\varrho }}{\left(1+| {D}_{i}u| )}^{{p}_{i}}{\rm{d}}z\right)}^{\tfrac{{p}_{i}-2}{{p}_{i}}}{\left(\mathop{\displaystyle \int }\limits_{{Q}_{2\varrho }}{| {D}_{s}u| }^{\sigma }{\rm{d}}z\right)}^{\tfrac{2{a}_{i}}{\sigma {p}_{1}}}{| h| }^{\tfrac{2{a}_{i}}{{p}_{1}}}{\left(\mathop{\displaystyle \int }\limits_{{Q}_{\tfrac{{\varrho }_{2}+{\varrho }_{1}}{2}}}{| {\tau }_{s,h}u| }^{\frac{2{q}_{3}\left({p}_{1}-{a}_{i})}{{p}_{1}}}{\rm{d}}z\right)}^{\tfrac{1}{{q}_{3}}}\\ & \le & C{| h| }^{\tfrac{2{a}_{i}}{{p}_{1}}}{\left(\mathop{\displaystyle \int }\limits_{{Q}_{\tfrac{{\varrho }_{2}+{\varrho }_{1}}{2}}}{| {\tau }_{s,h}u| }^{\frac{2{q}_{3}\left({p}_{1}-{a}_{i})}{{p}_{1}}}{\rm{d}}z\right)}^{\tfrac{1}{{q}_{3}}},\end{array}where CCdepends on p1,pi,ai,‖Diu‖Lpi(ΩT){p}_{1},{p}_{i},{a}_{i},{\Vert {D}_{i}u\Vert }_{{L}^{{p}_{i}}\left({\Omega }_{T})}, and ‖Dsu‖Lσ(Q2ϱ){\Vert {D}_{s}u\Vert }_{{L}^{\sigma }\left({Q}_{2\varrho })}, but not on hh. To estimate the remaining integral, we want to apply the parabolic Sobolev embedding from Lemma 2.3. For this, we need to choose ai{a}_{i}in such a way that (4.4)2q3(p1−ai)p1=σ(n+2)n⇔ai=(n+2)σp1−np1pi2pi.\frac{2{q}_{3}\left({p}_{1}-{a}_{i})}{{p}_{1}}=\frac{\sigma \left(n+2)}{n}\hspace{0.33em}\iff \hspace{0.33em}{a}_{i}=\frac{\left(n+2)\sigma {p}_{1}-n{p}_{1}{p}_{i}}{2{p}_{i}}.We have to check that this choice of ai{a}_{i}satisfies the condition (4.3). The upper bound ai<σp1pi{a}_{i}\lt \frac{\sigma {p}_{1}}{{p}_{i}}is satisfied due to pi>σ{p}_{i}\gt \sigma . To verify that the lower bound from (4.3) is satisfied, we use (1.5) to obtain: (n+2)σp1−np1pi≥p1((n+2)p1−npn)>p1(n+2)p1−n(n+2)p1n=0,\left(n+2)\sigma {p}_{1}-n{p}_{1}{p}_{i}\ge {p}_{1}\left(\left(n+2){p}_{1}-n{p}_{n})\gt {p}_{1}\left(\left(n+2){p}_{1}-n\frac{\left(n+2){p}_{1}}{n}\right)=0,and hence, ai>0{a}_{i}\gt 0. Thus, we have proved that (4.4) is an admissible choice for the parameter ai{a}_{i}. We can now apply Lemma 2.3 with (ϱ2,ϱ2+ϱ12)\left({\varrho }_{2},\frac{{\varrho }_{2}+{\varrho }_{1}}{2})instead of (ϱ,r)\left(\varrho ,r)to obtain ∫Qϱ2+ϱ12∣τs,hu∣2q3(p1−ai)p1dz1q3=∫Qϱ2+ϱ12∣τs,hu∣σ(n+2)ndznσ−npi≤C∫Qϱ2∣τs,hDu∣σ+τs,huϱ2−ϱ1σdznσ−npisupt∈(−ϱ22,0)∫Bϱ2∣τs,hu(⋅,t)∣2dx1−σpi≤C(N,n,σ,pi)(ϱ2−ϱ1)n(1−σpi)∫Q2ϱ∣Du∣σdznσ−npisupt∈(−ϱ22,0)∫Bϱ2∣τs,hu(⋅,t)∣2dx1−σpi≤C(N,n,σ,pi,‖Du‖Lσ(Q2ϱ))(ϱ2−ϱ1)n(1−σpi)supt∈(−ϱ22,0)∫Bϱ2∣τs,hu(⋅,t)∣2dx1−σpi.\begin{array}{rcl}{\left(\mathop{\displaystyle \int }\limits_{{Q}_{\tfrac{{\varrho }_{2}+{\varrho }_{1}}{2}}}{| {\tau }_{s,h}u| }^{\frac{2{q}_{3}\left({p}_{1}-{a}_{i})}{{p}_{1}}}{\rm{d}}z\right)}^{\tfrac{1}{{q}_{3}}}& =& {\left(\mathop{\displaystyle \int }\limits_{{Q}_{\tfrac{{\varrho }_{2}+{\varrho }_{1}}{2}}}{| {\tau }_{s,h}u| }^{\frac{\sigma \left(n+2)}{n}}{\rm{d}}z\right)}^{\tfrac{n}{\sigma }-\tfrac{n}{{p}_{i}}}\\ & \le & C{\left(\mathop{\displaystyle \int }\limits_{{Q}_{{\varrho }_{2}}}{| {\tau }_{s,h}Du| }^{\sigma }+{\left|\frac{{\tau }_{s,h}u}{{\varrho }_{2}-{\varrho }_{1}}\right|}^{\sigma }{\rm{d}}z\right)}^{\tfrac{n}{\sigma }-\tfrac{n}{{p}_{i}}}{\left(\mathop{\sup }\limits_{t\in \left(-{\varrho }_{2}^{2},0)}\mathop{\displaystyle \int }\limits_{{B}_{{\varrho }_{2}}}{| {\tau }_{s,h}u\left(\cdot ,t)| }^{2}{\rm{d}}x\right)}^{1-\tfrac{\sigma }{{p}_{i}}}\\ & \le & \frac{C\left(N,n,\sigma ,{p}_{i})}{{\left({\varrho }_{2}-{\varrho }_{1})}^{n\left(1-\tfrac{\sigma }{{p}_{i}})}}{\left(\mathop{\displaystyle \int }\limits_{{Q}_{2\varrho }}{| Du| }^{\sigma }{\rm{d}}z\right)}^{\tfrac{n}{\sigma }-\tfrac{n}{{p}_{i}}}{\left(\mathop{\sup }\limits_{t\in \left(-{\varrho }_{2}^{2},0)}\mathop{\displaystyle \int }\limits_{{B}_{{\varrho }_{2}}}{| {\tau }_{s,h}u\left(\cdot ,t)| }^{2}{\rm{d}}x\right)}^{1-\tfrac{\sigma }{{p}_{i}}}\\ & \le & \frac{C(N,n,\sigma ,{p}_{i},{\Vert Du\Vert }_{{L}^{\sigma }\left({Q}_{2\varrho })})}{{\left({\varrho }_{2}-{\varrho }_{1})}^{n\left(1-\tfrac{\sigma }{{p}_{i}})}}{\left(\mathop{\sup }\limits_{t\in \left(-{\varrho }_{2}^{2},0)}\mathop{\displaystyle \int }\limits_{{B}_{{\varrho }_{2}}}{| {\tau }_{s,h}u\left(\cdot ,t)| }^{2}{\rm{d}}x\right)}^{1-\tfrac{\sigma }{{p}_{i}}}.\end{array}In these calculations, we used the fact that ϱ2−ϱ1≤ϱ2<1{\varrho }_{2}-{\varrho }_{1}\le \frac{\varrho }{2}\lt 1and ∫Qϱ2∣τs,hDu∣σ+∣τs,hu∣σdz≤(2σ+∣h∣σ)∫Q2ϱ∣Du∣σdz≤(2σ+1)∫Q2ϱ∣Du∣σdz.\mathop{\int }\limits_{{Q}_{{\varrho }_{2}}}{| {\tau }_{s,h}Du| }^{\sigma }+{| {\tau }_{s,h}u| }^{\sigma }{\rm{d}}z\le \left({2}^{\sigma }+{| h| }^{\sigma })\mathop{\int }\limits_{{Q}_{2\varrho }}{| Du| }^{\sigma }{\rm{d}}z\le \left({2}^{\sigma }+1)\mathop{\int }\limits_{{Q}_{2\varrho }}{| Du| }^{\sigma }{\rm{d}}z.We combine the previous estimates to obtain RHSi≤C(ϱ2−ϱ1)n(1−σpi)∣h∣2aip1supt∈(−ϱ22,0)∫Bϱ2∣τs,hu(⋅,t)∣2dx1−σpi=C(ϱ2−ϱ1)n(1−σpi)∣h∣(n+2)σ−npipisupt∈(−ϱ22,0)∫Bϱ2∣τs,hu(⋅,t)∣2dx1−σpi,\begin{array}{rcl}{\text{RHS}}_{i}& \le & \frac{C}{{\left({\varrho }_{2}-{\varrho }_{1})}^{n\left(1-\tfrac{\sigma }{{p}_{i}})}}{| h| }^{\tfrac{2{a}_{i}}{{p}_{1}}}{\left(\mathop{\sup }\limits_{t\in \left(-{\varrho }_{2}^{2},0)}\mathop{\displaystyle \int }\limits_{{B}_{{\varrho }_{2}}}{| {\tau }_{s,h}u\left(\cdot ,t)| }^{2}{\rm{d}}x\right)}^{1-\tfrac{\sigma }{{p}_{i}}}\\ & =& \frac{C}{{\left({\varrho }_{2}-{\varrho }_{1})}^{n\left(1-\tfrac{\sigma }{{p}_{i}})}}{| h| }^{\tfrac{\left(n+2)\sigma -n{p}_{i}}{{p}_{i}}}{\left(\mathop{\sup }\limits_{t\in \left(-{\varrho }_{2}^{2},0)}\mathop{\displaystyle \int }\limits_{{B}_{{\varrho }_{2}}}{| {\tau }_{s,h}u\left(\cdot ,t)| }^{2}{\rm{d}}x\right)}^{1-\tfrac{\sigma }{{p}_{i}}},\end{array}where the constant CCdepends on N,n,σ,p1,pi,‖Diu‖Lpi(ΩT)N,n,\sigma ,{p}_{1},{p}_{i},{\Vert {D}_{i}u\Vert }_{{L}^{{p}_{i}}\left({\Omega }_{T})}, and ‖Du‖Lσ(Q2ϱ){\Vert Du\Vert }_{{L}^{\sigma }\left({Q}_{2\varrho })}. We multiply this inequality with 1(ϱ2−ϱ1)2\frac{1}{{\left({\varrho }_{2}-{\varrho }_{1})}^{2}}and apply Young’s inequality with exponents pipi−σ,piσ\frac{{p}_{i}}{{p}_{i}-\sigma },\frac{{p}_{i}}{\sigma }to obtain RHSi(ϱ2−ϱ1)2≤C(ϱ2−ϱ1)n+2−nσpi∣h∣(n+2)σ−npipisupt∈(−ϱ22,0)∫Bϱ2∣τs,hu(⋅,t)∣2dx1−σpi≤12nsupt∈(−ϱ22,0)∫Bϱ2∣τs,hu(⋅,t)∣2dx+C(ϱ2−ϱ1)(n+2)piσ−n∣h∣(n+2)σ−npiσ≤12nsupt∈(−ϱ22,0)∫Bϱ2∣τs,hu(⋅,t)∣2dx+C(ϱ2−ϱ1)4+4n∣h∣(n+2)σ−npiσ,\begin{array}{rcl}\frac{{\text{RHS}}_{i}}{{\left({\varrho }_{2}-{\varrho }_{1})}^{2}}& \le & \frac{C}{{\left({\varrho }_{2}-{\varrho }_{1})}^{n+2-\tfrac{n\sigma }{{p}_{i}}}}{| h| }^{\tfrac{\left(n+2)\sigma -n{p}_{i}}{{p}_{i}}}{\left(\mathop{\sup }\limits_{t\in \left(-{\varrho }_{2}^{2},0)}\mathop{\displaystyle \int }\limits_{{B}_{{\varrho }_{2}}}{| {\tau }_{s,h}u\left(\cdot ,t)| }^{2}{\rm{d}}x\right)}^{1-\tfrac{\sigma }{{p}_{i}}}\\ & \le & \frac{1}{2n}\mathop{\sup }\limits_{t\in \left(-{\varrho }_{2}^{2},0)}\mathop{\displaystyle \int }\limits_{{B}_{{\varrho }_{2}}}{| {\tau }_{s,h}u\left(\cdot ,t)| }^{2}{\rm{d}}x+\frac{C}{{\left({\varrho }_{2}-{\varrho }_{1})}^{\left(n+2)\tfrac{{p}_{i}}{\sigma }-n}}{| h| }^{\tfrac{\left(n+2)\sigma -n{p}_{i}}{\sigma }}\\ & \le & \frac{1}{2n}\mathop{\sup }\limits_{t\in \left(-{\varrho }_{2}^{2},0)}\mathop{\displaystyle \int }\limits_{{B}_{{\varrho }_{2}}}{| {\tau }_{s,h}u\left(\cdot ,t)| }^{2}{\rm{d}}x+\frac{C}{{\left({\varrho }_{2}-{\varrho }_{1})}^{4+\tfrac{4}{n}}}{| h| }^{\tfrac{\left(n+2)\sigma -n{p}_{i}}{\sigma }},\end{array}where CCadmits the same dependencies as specified earlier. In the last step, we used that ϱ2−ϱ1≤ϱ2<1{\varrho }_{2}-{\varrho }_{1}\le \frac{\varrho }{2}\lt 1and piσ≤pnp1<n+2n\frac{{p}_{i}}{\sigma }\le \frac{{p}_{n}}{{p}_{1}}\lt \frac{n+2}{n}due to (1.5). By inserting (4.2) and the last inequality into (4.1) and summing over i=1,…,ni=1,\ldots ,n, we obtain supt∈(−ϱ12,0)∫Bϱ1∣τs,hu(⋅,t)∣2dx+∑i=1n∫Qϱ1∣τs,hDiu∣pidz≤12supt∈(−ϱ22,0)∫Bϱ2∣τs,hu(⋅,t)∣2dx+C(ϱ2−ϱ1)4+4n∑i=1n∣h∣min2,(n+2)σ−npiσ≤12supt∈(−ϱ22,0)∫Bϱ2∣τs,hu(⋅,t)∣2dx+C(ϱ2−ϱ1)4+4n∣h∣(n+2)σ−npnσ.\begin{array}{l}\mathop{\sup }\limits_{t\in \left(-{\varrho }_{1}^{2},0)}\mathop{\displaystyle \int }\limits_{{B}_{{\varrho }_{1}}}{| {\tau }_{s,h}u\left(\cdot ,t)| }^{2}{\rm{d}}x+\mathop{\displaystyle \sum }\limits_{i=1}^{n}\mathop{\displaystyle \int }\limits_{{Q}_{{\varrho }_{1}}}{| {\tau }_{s,h}{D}_{i}u| }^{{p}_{i}}{\rm{d}}z\\ \hspace{1.0em}\le \frac{1}{2}\mathop{\sup }\limits_{t\in \left(-{\varrho }_{2}^{2},0)}\mathop{\displaystyle \int }\limits_{{B}_{{\varrho }_{2}}}{| {\tau }_{s,h}u\left(\cdot ,t)| }^{2}{\rm{d}}x+\frac{C}{{\left({\varrho }_{2}-{\varrho }_{1})}^{4+\tfrac{4}{n}}}\mathop{\displaystyle \sum }\limits_{i=1}^{n}{| h| }^{\min \left\{2,\tfrac{\left(n+2)\sigma -n{p}_{i}}{\sigma }\right\}}\\ \hspace{1.0em}\le \frac{1}{2}\mathop{\sup }\limits_{t\in \left(-{\varrho }_{2}^{2},0)}\mathop{\displaystyle \int }\limits_{{B}_{{\varrho }_{2}}}{| {\tau }_{s,h}u\left(\cdot ,t)| }^{2}{\rm{d}}x+\frac{C}{{\left({\varrho }_{2}-{\varrho }_{1})}^{4+\tfrac{4}{n}}}{| h| }^{\tfrac{\left(n+2)\sigma -n{p}_{n}}{\sigma }}.\end{array}At this point, we can apply Lemma 2.1 with the choices Φ(r)=supt∈(−r2,0)∫Br∣τs,hu(⋅,t)∣2dx+∑i=1n∫Qr∣τs,hDiu∣pidz,A=C∣h∣(n+2)σ−npnσ\begin{array}{rcl}\Phi \left(r)& =& \mathop{\sup }\limits_{t\in \left(-{r}^{2},0)}\mathop{\displaystyle \int }\limits_{{B}_{r}}{| {\tau }_{s,h}u\left(\cdot ,t)| }^{2}{\rm{d}}x+\mathop{\displaystyle \sum }\limits_{i=1}^{n}\mathop{\displaystyle \int }\limits_{{Q}_{r}}{| {\tau }_{s,h}{D}_{i}u| }^{{p}_{i}}{\rm{d}}z,\\ A& =& C{| h| }^{\tfrac{\left(n+2)\sigma -n{p}_{n}}{\sigma }}\end{array}and α=4+4n\alpha =4+\frac{4}{n}to absorb the sup\sup -term into the left-hand side. Thus, we conclude with supt∈(−(ϱ/2)2,0)∫Bϱ/2∣τs,hu(⋅,t)∣2dx+∑i=1n∫Qϱ/2∣τs,hDiu∣pidz≤Cϱ4+4n∣h∣(n+2)σ−npnσ,\mathop{\sup }\limits_{t\in (-{\left(\varrho \text{/}2)}^{2},0)}\mathop{\int }\limits_{{B}_{\varrho \text{/}2}}{| {\tau }_{s,h}u\left(\cdot ,t)| }^{2}{\rm{d}}x+\mathop{\sum }\limits_{i=1}^{n}\mathop{\int }\limits_{{Q}_{\varrho \text{/}2}}{| {\tau }_{s,h}{D}_{i}u| }^{{p}_{i}}{\rm{d}}z\le \frac{C}{{\varrho }^{4+\tfrac{4}{n}}}{| h| }^{\tfrac{\left(n+2)\sigma -n{p}_{n}}{\sigma }},with a constant CCthat does not depend on hh. We also note that (n+2)σ−npnσ≥(n+2)p1−npnσ>0\frac{\left(n+2)\sigma -n{p}_{n}}{\sigma }\ge \frac{\left(n+2){p}_{1}-n{p}_{n}}{\sigma }\gt 0due to (1.5).Step 2: Let us summarize the results from Step 1. Under the assumption that Q2ϱ⊂ΩT{Q}_{2\varrho }\subset {\Omega }_{T}and Du∈Llocσ(ΩT,RN×n)for someσ∈[p1,pn),Du\in {L}_{\hspace{0.1em}\text{loc}\hspace{0.1em}}^{\sigma }({\Omega }_{T},{{\mathbb{R}}}^{N\times n})\hspace{1.0em}\hspace{0.1em}\text{for some}\hspace{0.1em}\hspace{0.33em}\sigma \in \left[{p}_{1},{p}_{n}),we proved that there exists a constant CC(independent of hh) such that for all ∣h∣<ϱ| h| \lt \varrho , we have supt∈(−(ϱ/2)2,0)∫Bϱ/2∣τs,hu(⋅,t)∣2dx+∑i=1n∫Qϱ/2∣τs,hDiu∣pidz≤Cϱ4+4n∣h∣(n+2)σ−npnσ.\mathop{\sup }\limits_{t\in (-{\left(\varrho \text{/}2)}^{2},0)}\mathop{\int }\limits_{{B}_{\varrho \text{/}2}}{| {\tau }_{s,h}u\left(\cdot ,t)| }^{2}{\rm{d}}x+\mathop{\sum }\limits_{i=1}^{n}\mathop{\int }\limits_{{Q}_{\varrho \text{/}2}}{| {\tau }_{s,h}{D}_{i}u| }^{{p}_{i}}{\rm{d}}z\le \frac{C}{{\varrho }^{4+\tfrac{4}{n}}}{| h| }^{\tfrac{\left(n+2)\sigma -n{p}_{n}}{\sigma }}.We are now in the position to apply Lemma 2.12, which asserts that uulies in the following fractional Sobolev spaces for every O⋐Bϱ/2{\mathcal{O}}\hspace{0.33em}\Subset \hspace{0.33em}{B}_{\varrho \text{/}2}: u∈L∞(−(ϱ/2)2,0;Wμ,2(O,RN))∀μ∈(0,μmax),μmax≔(n+2)σ−npn2σu\in {L}^{\infty }(-{\left(\varrho \text{/}2)}^{2},0;\hspace{0.33em}{W}^{\mu ,2}({\mathcal{O}},{{\mathbb{R}}}^{N}))\hspace{1.0em}\forall \mu \in (0,{\mu }_{\max }),\hspace{0.33em}{\mu }_{\max }:= \frac{\left(n+2)\sigma -n{p}_{n}}{2\sigma }and Diu∈Lpi(−(ϱ/2)2,0;Wλ,pi(O,RN))∀λ∈(0,λmax(i)),λmax(i)≔(n+2)σ−npnσpi.{D}_{i}u\in {L}^{{p}_{i}}(-{\left(\varrho \text{/}2)}^{2},0;\hspace{0.33em}{W}^{\lambda ,{p}_{i}}({\mathcal{O}},{{\mathbb{R}}}^{N}))\hspace{1.0em}\forall \lambda \in (0,{\lambda }_{\max }^{\left(i)}),\hspace{0.33em}{\lambda }_{\max }^{\left(i)}:= \frac{\left(n+2)\sigma -n{p}_{n}}{\sigma {p}_{i}}.Since u∈Lp1(−(ϱ/2)2,0;W1,p1(O,RN))u\in {L}^{{p}_{1}}(-{\left(\varrho \text{/}2)}^{2},0;\hspace{0.33em}{W}^{1,{p}_{1}}({\mathcal{O}},{{\mathbb{R}}}^{N}))by definition, we can apply Lemma 2.11 with i=1i=1to obtain D1u∈Ls(O×(−(ϱ/2)2,0),RN){D}_{1}u\in {L}^{s}({\mathcal{O}}\times \left(-{\left(\varrho \text{/}2)}^{2},0),{{\mathbb{R}}}^{N})for all s>p1s\gt {p}_{1}such that (s−p1)1−μmax+n2<λmax(1)p1⇔(s−p1)1−(n+2)σ−npn2σ+n2<(n+2)σ−npnσ⇔(s−p1)npn2σ<(n+2)σ−npnσ⇔s<p1+2(n+2)σ−2npnnpn.\begin{array}{l}\left(s-{p}_{1})\left(1-{\mu }_{\max }+\frac{n}{2}\right)\lt {\lambda }_{\max }^{\left(1)}{p}_{1}\\ \hspace{1.0em}\iff \left(s-{p}_{1})\left(1-\frac{\left(n+2)\sigma -n{p}_{n}}{2\sigma }+\frac{n}{2}\right)\lt \frac{\left(n+2)\sigma -n{p}_{n}}{\sigma }\\ \hspace{1.0em}\iff \left(s-{p}_{1})\frac{n{p}_{n}}{2\sigma }\lt \frac{\left(n+2)\sigma -n{p}_{n}}{\sigma }\\ \hspace{1.0em}\iff s\lt {p}_{1}+\frac{2\left(n+2)\sigma -2n{p}_{n}}{n{p}_{n}}.\end{array}Since the center of the cylinder Qϱ{Q}_{\varrho }was arbitrary, we thus obtain (4.5)D1u∈Llocs(ΩT,RN)∀s<sˆ1(σ)≔p1+2(n+2)σ−2npnnpn.{D}_{1}u\in {L}_{\hspace{0.1em}\text{loc}\hspace{0.1em}}^{s}({\Omega }_{T},{{\mathbb{R}}}^{N})\hspace{1.0em}\forall s\lt {\hat{s}}_{1}\left(\sigma ):= {p}_{1}+\frac{2\left(n+2)\sigma -2n{p}_{n}}{n{p}_{n}}.Step 3: Let us summarize the previous two steps: From the assumption that Du∈Llocσ(ΩT,RN×n)Du\in {L}_{\hspace{0.1em}\text{loc}\hspace{0.1em}}^{\sigma }({\Omega }_{T},{{\mathbb{R}}}^{N\times n})for some σ∈[p1,pn)\sigma \in \left[{p}_{1},{p}_{n}), we have deduced the improved integrability (4.5). We will now show that (4.6)Sˆ≔infσ∈[p1,pn)(sˆ1(σ)−σ)>0.\hat{S}:= \mathop{\inf }\limits_{\sigma \in \left[{p}_{1},{p}_{n})}({\hat{s}}_{1}\left(\sigma )-\sigma )\gt 0.We have sˆ1(σ)−σ=p1+(2(n+2)−npn)σ−2npnnpn.{\hat{s}}_{1}\left(\sigma )-\sigma ={p}_{1}+\frac{\left(2\left(n+2)-n{p}_{n})\sigma -2n{p}_{n}}{n{p}_{n}}.There are three cases to consider (depending on the sign of 2(n+2)−npn2\left(n+2)-n{p}_{n}).Case 1: If 2(n+2)−npn>0,2\left(n+2)-n{p}_{n}\gt 0,we have infσ∈[p1,pn)(sˆ1(σ)−σ)=sˆ1(p1)−p1=2(n+2)p1−2npnnpn>0,\mathop{\inf }\limits_{\sigma \in \left[{p}_{1},{p}_{n})}({\hat{s}}_{1}\left(\sigma )-\sigma )={\hat{s}}_{1}\left({p}_{1})-{p}_{1}=\frac{2\left(n+2){p}_{1}-2n{p}_{n}}{n{p}_{n}}\gt 0,due to (1.5).Case 2: If 2(n+2)−npn=0⇔pn=2(n+2)n=2+4n,2\left(n+2)-n{p}_{n}=0\hspace{0.33em}\iff \hspace{0.33em}{p}_{n}=\frac{2\left(n+2)}{n}=2+\frac{4}{n},we have p1>2{p}_{1}\gt 2, since otherwise the gap condition (1.4) would not hold. Hence, we have infσ∈[p1,pn)(sˆ1(σ)−σ)=sˆ1(p1)−p1=p1−2>0.\mathop{\inf }\limits_{\sigma \in \left[{p}_{1},{p}_{n})}({\hat{s}}_{1}\left(\sigma )-\sigma )={\hat{s}}_{1}\left({p}_{1})-{p}_{1}={p}_{1}-2\gt 0.Case 3: If 2(n+2)−npn<0,2\left(n+2)-n{p}_{n}\lt 0,we have infσ∈[p1,pn)(sˆ1(σ)−σ)=sˆ1(pn)−pn=p1+4n−pn>0,\mathop{\inf }\limits_{\sigma \in \left[{p}_{1},{p}_{n})}({\hat{s}}_{1}\left(\sigma )-\sigma )={\hat{s}}_{1}\left({p}_{n})-{p}_{n}={p}_{1}+\frac{4}{n}-{p}_{n}\gt 0,due to (1.4). Thus, we can conclude that (4.6) holds in all cases.Step 4: Since the integrability gain from σ\sigma to sˆ1(σ){\hat{s}}_{1}\left(\sigma )is uniform with respect to σ\sigma due to (4.6), we can now perform an iteration procedure to show the desired higher integrability. First, we want to show that D1u∈Llocp2(ΩT,RN){D}_{1}u\in {L}_{{\rm{loc}}}^{{p}_{2}}({\Omega }_{T},{{\mathbb{R}}}^{N}). To achieve this, we consider the following iteration scheme (which is possible since Du∈Llocp1(ΩT,RN×n)Du\in {L}_{{\rm{loc}}}^{{p}_{1}}({\Omega }_{T},{{\mathbb{R}}}^{N\times n})by definition): σ1=p1⇒D1u∈Llocp1+Sˆ⇒Du∈Llocmin{p1+Sˆ,p2}σ2=p1+Sˆ⇒D1u∈Llocp1+2Sˆ⇒Du∈Llocmin{p1+2Sˆ,p2}⋮\begin{array}{rclllll}{\sigma }_{1}& =& {p}_{1}& \Rightarrow & {D}_{1}u\in {L}_{{\rm{loc}}}^{{p}_{1}+\hat{S}}& \Rightarrow & Du\in {L}_{{\rm{loc}}}^{\min \{{p}_{1}+\hat{S},{p}_{2}\}}\\ {\sigma }_{2}& =& {p}_{1}+\hat{S}& \Rightarrow & {D}_{1}u\in {L}_{{\rm{loc}}}^{{p}_{1}+2\hat{S}}& \Rightarrow & Du\in {L}_{{\rm{loc}}}^{\min \{{p}_{1}+2\hat{S},{p}_{2}\}}\\ & \vdots & & & & & \end{array}Clearly, we obtain D1u∈Llocp2(ΩT,RN){D}_{1}u\in {L}_{{\rm{loc}}}^{{p}_{2}}\left({\Omega }_{T},{{\mathbb{R}}}^{N}), and hence, Du∈Llocp2(ΩT,RN×n)Du\in {L}_{{\rm{loc}}}^{{p}_{2}}\left({\Omega }_{T},{{\mathbb{R}}}^{N\times n})after finitely many iterations. The next step in the iteration is to show that D1u,D2u∈Llocp3{D}_{1}u,{D}_{2}u\in {L}_{{\rm{loc}}}^{{p}_{3}}. To achieve this, we apply Lemma (2.11) with p2{p}_{2}instead of ppto obtain that D2u∈Llocs(ΩT,RN){D}_{2}u\in {L}_{{\rm{loc}}}^{s}({\Omega }_{T},{{\mathbb{R}}}^{N})for all s>p2s\gt {p}_{2}such that (s−p2)1−μmax+n2<λmax(2)p2⇔s<p2+2(n+2)σ−2npnnpn≕sˆ2(σ).\left(s-{p}_{2})\left(1-{\mu }_{\max }+\frac{n}{2}\right)\lt {\lambda }_{\max }^{\left(2)}{p}_{2}\iff s\lt {p}_{2}+\frac{2\left(n+2)\sigma -2n{p}_{n}}{n{p}_{n}}\hspace{0.33em}=: \hspace{0.33em}{\hat{s}}_{2}\left(\sigma ).Due to p2≥p1{p}_{2}\ge {p}_{1}, we have sˆ2(σ)−σ≥sˆ1(σ)−σ≥Sˆ{\hat{s}}_{2}\left(\sigma )-\sigma \ge {\hat{s}}_{1}\left(\sigma )-\sigma \ge \hat{S}for all σ∈[p2,pn)\sigma \in \left[{p}_{2},{p}_{n}), which means that we gain at least as much integrability for D2u{D}_{2}uas for D1u{D}_{1}u. We can now use the following iteration scheme to simultaneously improve the integrability for D1u{D}_{1}uand D2u{D}_{2}u: σ1=p2⇒D1u,D2u∈Llocp2+Sˆ⇒Du∈Llocmin{p2+Sˆ,p3}σ2=p2+Sˆ⇒D1u,D2u∈Llocp2+2Sˆ⇒Du∈Llocmin{p2+2Sˆ,p3}⋮\begin{array}{rclllll}{\sigma }_{1}& =& {p}_{2}& \Rightarrow & {D}_{1}u,{D}_{2}u\in {L}_{{\rm{loc}}}^{{p}_{2}+\hat{S}}& \Rightarrow & Du\in {L}_{{\rm{loc}}}^{\min \{{p}_{2}+\hat{S},{p}_{3}\}}\\ {\sigma }_{2}& =& {p}_{2}+\hat{S}& \Rightarrow & {D}_{1}u,{D}_{2}u\in {L}_{{\rm{loc}}}^{{p}_{2}+2\hat{S}}& \Rightarrow & Du\in {L}_{{\rm{loc}}}^{\min \{{p}_{2}+2\hat{S},{p}_{3}\}}\\ & \vdots & & & & & \end{array}After finitely many steps of this iteration, we obtain Du∈Llocp3(ΩT,RN×n)Du\in {L}_{{\rm{loc}}}^{{p}_{3}}({\Omega }_{T},{{\mathbb{R}}}^{N\times n}). In an analogous way, we can improve the integrability of D1u,D2u,D3u{D}_{1}u,{D}_{2}u,{D}_{3}uto obtain Du∈Llocp4(ΩT,RN×n)Du\in {L}_{{\rm{loc}}}^{{p}_{4}}({\Omega }_{T},{{\mathbb{R}}}^{N\times n}). Inductively we obtain Du∈Llocpn(ΩT,RN×n)Du\in {L}_{{\rm{loc}}}^{{p}_{n}}({\Omega }_{T},{{\mathbb{R}}}^{N\times n}). Due to (4.5), the maximum amount of integrability that can be obtained for D1u{D}_{1}u(and hence for DuDu) is limited by limσ→pnsˆ1(σ)=p1+4n.\mathop{\mathrm{lim}}\limits_{\sigma \to {p}_{n}}{\hat{s}}_{1}\left(\sigma )={p}_{1}+\frac{4}{n}.This implies that we obtain Du∈LlocsDu\in {L}_{{\rm{loc}}}^{s}for all s<p1+4ns\lt {p}_{1}+\frac{4}{n}. This proves the assertion of the theorem.□ http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Nonlinear Analysis de Gruyter

Higher integrability for anisotropic parabolic systems of p-Laplace type

Advances in Nonlinear Analysis , Volume 12 (1): 1 – Jan 1, 2023

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de Gruyter
Copyright
© 2023 the author(s), published by De Gruyter
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2191-950X
DOI
10.1515/anona-2022-0308
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Abstract

1IntroductionThe subject of this article are parabolic systems (1.1)ut−divDf(Du)=0inΩT=Ω×(0,T),{u}_{t}-{\rm{div}}Df\left(Du)=0\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{\Omega }_{T}=\Omega \times \left(0,T),where Ω⊂Rn(n≥2)\Omega \subset {{\mathbb{R}}}^{n}\hspace{0.33em}\left(n\ge 2)is a bounded domain (i.e., a nonempty, open and connected subset of Rn{{\mathbb{R}}}^{n}) and u:ΩT→RN(N≥1)u:{\Omega }_{T}\to {{\mathbb{R}}}^{N}\hspace{0.33em}\left(N\ge 1)is the desired function. By ut,Du{u}_{t},Du, we denote the derivative with respect to time respectively the spatial gradient of uu. We assume that f∈C2(RN×n)f\in {C}^{2}({{\mathbb{R}}}^{N\times n})satisfies a certain nonstandard growth condition. A very general class of equations with a nonstandard growth condition is given by the so-called pp–qqgrowth condition ∣z∣p≤f(ξ)≤L(1+∣ξ∣q){| z| }^{p}\le f\left(\xi )\le L\left(1+{| \xi | }^{q})with exponents 1<p<q1\lt p\lt q. A typical example for a function that fulfills such a condition is f(ξ)=1p∣ξ∣p+1q∣ξn∣qf\left(\xi )=\frac{1}{p}{| \xi | }^{p}+\frac{1}{q}{| {\xi }_{n}| }^{q}. In the elliptic setting, such equations, respectively the corresponding variational problem, min∫Ωf(Du)dx,\min \mathop{\int }\limits_{\Omega }f\left(Du){\rm{d}}x,were first examined in the late 1980s, starting with the seminal papers by Marcellini. In [19,25], Giaquinta and Marcellini gave counterexamples to the local Hölder continuity of minimizers (which was already known for the standard growth case p=qp=q), if the gap q−pq-pbetween the exponents is too large. Subsequently, for the scalar case N=1N=1, the local boundedness of the gradient of minimizers was proved by P. Marcellini under the condition that the exponents p,qp,qare not too far apart [26,27]. In the vectorial case N≥2N\ge 2, the partial Hölder continuity of weak solutions, i.e., Hölder continuity up to a subset with measure zero, was proved in [2]. There is a vast amount of literature on both elliptic and parabolic equations with pp–qqgrowth conditions, but we want to point out two recent articles in particular: In [9], the local boundedness of minimizers for functionals with anisotropic pp–qqgrowth conditions was established under sharp assumptions on the exponents, and [28] features a variational approach for a class of parabolic equations under very general growth conditions.In this article, however, we will not consider a general pp–qqgrowth condition but instead focus on a special case of anisotropic growth conditions. To make this precise, we assume that the integrand f∈C2(RN×n)f\in {C}^{2}({{\mathbb{R}}}^{N\times n})satisfies the following growth and ellipticity conditions: (1.2)∣f(ξ)∣≤L1+∑i=1n∣ξi∣pi∣D2f(ξ)∣≤L1+∑i=1n∣ξi∣pi−2⟨D2f(ξ)η,η⟩≥ν∑i=1n∣ξi∣pi−2∣ηi∣2\left\{\begin{array}{l}| f\left(\xi )| \le L\left(1+\mathop{\displaystyle \sum }\limits_{i=1}^{n}{| {\xi }_{i}| }^{{p}_{i}}\right)\hspace{1.0em}\\ | {D}^{2}f\left(\xi )| \le L\left(1+\mathop{\displaystyle \sum }\limits_{i=1}^{n}{| {\xi }_{i}| }^{{p}_{i}-2}\right)\hspace{1.0em}\\ \langle {D}^{2}f\left(\xi )\eta ,\eta \rangle \ge \nu \mathop{\displaystyle \sum }\limits_{i=1}^{n}{| {\xi }_{i}| }^{{p}_{i}-2}{| {\eta }_{i}| }^{2}\hspace{1.0em}\end{array}\right.for all ξ,η∈RN×n\xi ,\eta \in {{\mathbb{R}}}^{N\times n}with some constants 0<ν≤L0\lt \nu \le L. Here, we have denoted ξi=(ξi1,…,ξiN),ηi=(ηi1,…,ηiN)∈RN{\xi }_{i}=({\xi }_{i1},\ldots ,{\xi }_{iN}),{\eta }_{i}=({\eta }_{i1},\ldots ,{\eta }_{iN})\in {{\mathbb{R}}}^{N}. We also assume that the growth exponents p1,…,pn>1{p}_{1},\ldots ,{p}_{n}\gt 1satisfy min{pi}<max{pi}\min \{{p}_{i}\}\lt \max \{{p}_{i}\}. The prototype for such systems is the parabolic pi{p}_{i}-Laplace system ut−∑i=1n∂∂xi(∣Diu∣pi−2Diu)=0.{u}_{t}-\mathop{\sum }\limits_{i=1}^{n}\frac{\partial }{\partial {x}_{i}}({| {D}_{i}u| }^{{p}_{i}-2}{D}_{i}u)=0.This is precisely our system (1.1) for the choice f(ξ)=∑i=1n1pi∣ξi∣pi,f\left(\xi )=\mathop{\sum }\limits_{i=1}^{n}\frac{1}{{p}_{i}}{| {\xi }_{i}| }^{{p}_{i}},which obviously also satisfies the pp–qqgrowth condition with p=min{pi}p=\min \{{p}_{i}\}and q=max{pi}q=\max \{{p}_{i}\}. Without loss of generality, we can assume that the exponents are ordered, i.e., 1<p1≤p2≤…≤pn1\lt {p}_{1}\le {p}_{2}\le \ldots \le {p}_{n}, and hence pmin=p1,pmax=pn{p}_{\min }={p}_{1},{p}_{\max }={p}_{n}.Let us first mention a few important regularity results for elliptic equations satisfying the pi{p}_{i}-growth conditions (1.2), with no attempt at completeness. Similarly as for pp–qqgrowth conditions, the main feature of the theory is that the range of the exponents pi{p}_{i}must be sufficiently small for any regularity results to hold. The local boundedness of weak solutions under a sharp condition on the exponents pi{p}_{i}was proved in [18]. Based on this, the local Hölder continuity of bounded weak solutions in the special case p1=2{p}_{1}=2, p2=…=pn>2{p}_{2}=\ldots ={p}_{n}\gt 2was proved in [24]. This result was later extended to the case 1<p1<p2=…=pn1\lt {p}_{1}\lt {p}_{2}=\ldots ={p}_{n}in [13]. Recently, the local Hölder continuity of weak solutions was proved under the condition that pmax−pmin≤1c{p}_{\max }-{p}_{\min }\le \frac{1}{c}, with a constant ccthat depends only on the data [10]. In particular, this result allows all the exponents pi{p}_{i}to be different. In the superquadratic case 2≤p1≤…≤pn2\le {p}_{1}\hspace{0.33em}\le \ldots \le {p}_{n}, the local Lipschitz continuity of bounded weak solutions was proved in the very recent paper [4]. A sharp result about the speed of propagation for parabolic equations with pi{p}_{i}-growth can be found in [14].In this article, however, we will focus on the higher integrability of weak solutions in the parabolic setting. We want to show that the spatial derivatives Diu{D}_{i}u, which are a priori only in Lpi{L}^{{p}_{i}}, are in fact integrable to some higher power. In the elliptic setting, such higher integrability results for integrands with pp–qq-growth were proved in [15,16] and later refined in [8]. The elliptic result for the anisotropic growth conditions (1.2) can be found in [22]. In this article, we want to extend this result to the parabolic setting. For systems with a general pp–qq-growth condition, this has already been achieved in [6], where the higher integrability of the gradient was proved in the superquadratic case p≥2p\ge 2. The corresponding result for the subquadratic case p<2p\lt 2(and for an integrand ffwith (x,t)\left(x,t)-dependence) was proved in [31]. Furthermore, parabolic equations with pp–qq-growth and with (x,t)\left(x,t)-dependent coefficients were treated in [5]. Instead, we will only focus on the special growth conditions (1.2) in this article and restrict our attention to integrands ffthat depend only on the gradient variable. As will be discussed in Remark 1.5, in this special case, we can obtain a better bound on the gap between p1{p}_{1}and pn{p}_{n}than in the case of pp–qqgrowth conditions. Furthermore, we will only consider the superquadratic case, i.e., 2≤p1≤…≤pn2\le {p}_{1}\le \ldots \le {p}_{n}, and of course p1<pn{p}_{1}\lt {p}_{n}.To define weak solutions, we first need to introduce anisotropic versions of the classical Sobolev- and Bochner-spaces. Following Lions [23, Chapter 2.1.7], we denote Wxi1,pi(Ω,RN)={u∈Lpi(Ω,RN):Diu∈Lpi(Ω,RN)}{W}_{{x}_{i}}^{1,{p}_{i}}(\Omega ,{{\mathbb{R}}}^{N})=\{u\in {L}^{{p}_{i}}(\Omega ,{{\mathbb{R}}}^{N}):{D}_{i}u\in {L}^{{p}_{i}}(\Omega ,{{\mathbb{R}}}^{N})\}for a fixed index i∈{1,…,n}i\in \{1,\ldots ,n\}. Endowed with the norm ‖u‖Wxi1,pi≔‖u‖Lpi+‖Diu‖Lpi,{\Vert u\Vert }_{{W}_{{x}_{i}}^{1,{p}_{i}}}:= {\Vert u\Vert }_{{L}^{{p}_{i}}}+{\Vert {D}_{i}u\Vert }_{{L}^{{p}_{i}}},this space becomes a Banach space. Furthermore, we denote p=(p1,…,pn){\bf{p}}=\left({p}_{1},\ldots ,{p}_{n})and we define the anisotropic Sobolev space W1,p(Ω,RN){W}^{1,{\bf{p}}}(\Omega ,{{\mathbb{R}}}^{N})as follows: W1,p(Ω,RN)=⋂i=1nWxi1,pi(Ω,RN)={u∈Lpn(Ω,RN):Diu∈Lpi(Ω,RN)fori=1,…,n}.{W}^{1,{\bf{p}}}(\Omega ,{{\mathbb{R}}}^{N})=\mathop{\bigcap }\limits_{i=1}^{n}{W}_{{x}_{i}}^{1,{p}_{i}}(\Omega ,{{\mathbb{R}}}^{N})=\{u\in {L}^{{p}_{n}}(\Omega ,{{\mathbb{R}}}^{N}):{D}_{i}u\in {L}^{{p}_{i}}(\Omega ,{{\mathbb{R}}}^{N})\hspace{0.33em}\hspace{0.1em}\text{for}\hspace{0.1em}\hspace{0.33em}i=1,\ldots ,n\}.We equip this space with the norm ‖u‖W1,p(Ω,RN)≔‖u‖Lpn(Ω,RN)+∑i=1n‖Diu‖Lpi(Ω,RN).{\Vert u\Vert }_{{W}^{1,{\bf{p}}}\left(\Omega ,{{\mathbb{R}}}^{N})}:= {\Vert u\Vert }_{{L}^{{p}_{n}}\left(\Omega ,{{\mathbb{R}}}^{N})}+\mathop{\sum }\limits_{i=1}^{n}{\Vert {D}_{i}u\Vert }_{{L}^{{p}_{i}}\left(\Omega ,{{\mathbb{R}}}^{N})}.Now we will turn our attention to anisotropic Bochner spaces. For a fixed index i∈{1,…,n}i\in \{1,\ldots ,n\}, we denote by Lpi(0,T;Wxi1,pi(Ω,RN)){L}^{{p}_{i}}(0,T;\hspace{0.33em}{W}_{{x}_{i}}^{1,{p}_{i}}(\Omega ,{{\mathbb{R}}}^{N}))the “classical” Bochner space with respect to the variable xi{x}_{i}. A suitable anisotropic Bochner space, which is needed for the definition of weak solutions, is then defined in the following way.Definition 1.1The anisotropic Bochner space Lp(0,T;W1,p(Ω,RN)){L}^{{\bf{p}}}(0,T;\hspace{0.33em}{W}^{1,{\bf{p}}}(\Omega ,{{\mathbb{R}}}^{N}))is defined as follows: Lp(0,T;W1,p(Ω,RN))=⋂i=1nLpi(0,T;Wxi1,pi(Ω,RN)).{L}^{{\bf{p}}}(0,T;\hspace{0.33em}{W}^{1,{\bf{p}}}(\Omega ,{{\mathbb{R}}}^{N}))=\mathop{\bigcap }\limits_{i=1}^{n}{L}^{{p}_{i}}(0,T;\hspace{0.33em}{W}_{{x}_{i}}^{1,{p}_{i}}(\Omega ,{{\mathbb{R}}}^{N})).In particular, any function Lp(0,T;W1,p(Ω,RN)){L}^{{\bf{p}}}(0,T;\hspace{0.33em}{W}^{1,{\bf{p}}}(\Omega ,{{\mathbb{R}}}^{N}))has the following properties: (1)u(⋅,t)∈W1,p(Ω,RN)u\left(\cdot ,t)\in {W}^{1,{\bf{p}}}(\Omega ,{{\mathbb{R}}}^{N})for a.e. t∈(0,T)t\in \left(0,T)and the mapping (0,T)∋t↦u(⋅,t)∈W1,p(Ω,RN)\left(0,T)\hspace{0.33em}\ni \hspace{0.33em}t\mapsto u\left(\cdot ,t)\in {W}^{1,{\bf{p}}}(\Omega ,{{\mathbb{R}}}^{N})is strongly measurable.(2)The anisotropic Bochner norm ‖u‖≔∫0T‖u(⋅,t)‖Lpn(Ω,RN)pndt1pn+∑i=1n∫0T‖Diu(⋅,t)‖Lpi(Ω,RN)pidt1pi\Vert u\Vert := {\left(\underset{0}{\overset{T}{\int }}{\Vert u\left(\cdot ,t)\Vert }_{{L}^{{p}_{n}}\left(\Omega ,{{\mathbb{R}}}^{N})}^{{p}_{n}}{\rm{d}}t\right)}^{\tfrac{1}{{p}_{n}}}+\mathop{\sum }\limits_{i=1}^{n}{\left(\underset{0}{\overset{T}{\int }}{\Vert {D}_{i}u\left(\cdot ,t)\Vert }_{{L}^{{p}_{i}}\left(\Omega ,{{\mathbb{R}}}^{N})}^{{p}_{i}}{\rm{d}}t\right)}^{\tfrac{1}{{p}_{i}}}is finite.With this notion of anisotropic Bochner spaces at hand, we can define weak solutions for (1.1) in the following way.Definition 1.2A function u∈Lp(0,T;W1,p(Ω,RN))∩L∞(0,T;L2(Ω,RN))u\in {L}^{{\bf{p}}}(0,T;\hspace{0.33em}{W}^{1,{\bf{p}}}(\Omega ,{{\mathbb{R}}}^{N}))\cap {L}^{\infty }(0,T;\hspace{0.33em}{L}^{2}(\Omega ,{{\mathbb{R}}}^{N}))is called a weak solution of (1.1), if the weak formulation (1.3)∫ΩTu⋅φt−⟨Df(Du),Dφ⟩dz=0\mathop{\int }\limits_{{\Omega }_{T}}u\cdot {\varphi }_{t}-\langle Df\left(Du),D\varphi \rangle {\rm{d}}z=0holds for all test functions φ∈C0∞(ΩT,RN)\varphi \in {C}_{0}^{\infty }({\Omega }_{T},{{\mathbb{R}}}^{N}).Remark 1.3By “⋅<mml:mpadded xmlns:ali="http://www.niso.org/schemas/ali/1.0/"xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"voffset="-0.2ex">\cdot </mml:mpadded>”, we denote the scalar product on RN{{\mathbb{R}}}^{N}(or Rn{{\mathbb{R}}}^{n}), and by ⟨⋅,⋅⟩\langle \cdot ,\cdot \rangle , we denote the scalar product on RN×n{{\mathbb{R}}}^{N\times n}.The existence of weak solutions can be deduced from the theory of monotone operators, see [23, Chapter 2.1.7, Theorem 1.4]. In this article, the main focus lies instead on the (local) higher integrability of weak solutions, which can be used as a starting tool for further regularity results. Our main result is the following theorem.Theorem 1.4Let f∈C2(RN×n)f\in {C}^{2}({{\mathbb{R}}}^{N\times n})be a function satisfying the structure conditions (1.2). Furthermore, let us assume that the exponents pi{p}_{i}are ordered, i.e., p1≤p2≤…≤pn{p}_{1}\le {p}_{2}\le \ldots \le {p}_{n}and satisfy the gap condition(1.4)2≤p1<pn<p1+4n.2\le {p}_{1}\lt {p}_{n}\lt {p}_{1}+\frac{4}{n}.Let u be a weak solution of (1.1). Then we haveDu∈Llocs(ΩT,RN×n)∀s<p1+4n.Du\in {L}_{{\rm{loc}}}^{s}({\Omega }_{T},{{\mathbb{R}}}^{N\times n})\hspace{1.0em}\forall s\lt {p}_{1}+\frac{4}{n}.In other words, this theorem shows that DuDuis locally in Lpn+ε{L}^{{p}_{n}+\varepsilon }for some ε>0\varepsilon \gt 0. Of course, we can also use this result to improve the integrability of uu.Remark 1.5Let us compare the condition (1.4) with the corresponding elliptic condition from [22, Theorem A.1]. In the elliptic case, the exponents are required to satisfy the bound pn<np1n−2=p1+2p1n−2,{p}_{n}\lt \frac{n{p}_{1}}{n-2}={p}_{1}+\frac{2{p}_{1}}{n-2},while the parabolic bound can be rewritten as follows: pn<p1+4n=p1+2p1(n+2)−2⋅2p1.{p}_{n}\lt {p}_{1}+\frac{4}{n}={p}_{1}+\frac{2{p}_{1}}{\left(n+2)-2}\cdot \frac{2}{{p}_{1}}.This seems to be the natural bound since the dimension nnis replaced by n+2n+2(which is due to the typical parabolic scaling in time) and the parabolic deficit 2p1\frac{2}{{p}_{1}}shows up.We can also compare (1.4) to the corresponding condition for systems that satisfy a general pp–qq-growth condition. In the elliptic case, the higher integrability of DuDuwas proved under the condition that q−p<2pnq-p\lt \frac{2p}{n}[15, Theorem 2.1], while in the parabolic case, the gap condition is given by q−p<2pn+2⋅2p=4n+2q-p\lt \frac{2p}{n+2}\cdot \frac{2}{p}=\frac{4}{n+2}[6, Lemma 6.8], which is a more restrictive condition than (1.4). This is explained by the fact that for systems with a pp–qq-growth condition, the weak derivatives Diu{D}_{i}uare a priori only in Lp{L}^{p}. In our case, however, we know a priori that Diu{D}_{i}uis in Lpi{L}^{{p}_{i}}with pi≥p1{p}_{i}\ge {p}_{1}. This better integrability leads to a less restrictive condition on the gap pmax−pmin{p}_{\max }-{p}_{\min }.Remark 1.6For later use, we note that (1.4) implies (1.5)pn<p1+4n≤p1+2p1n=(n+2)p1n⇒pnp1<n+2n.{p}_{n}\lt {p}_{1}+\frac{4}{n}\le {p}_{1}+\frac{2{p}_{1}}{n}=\frac{\left(n+2){p}_{1}}{n}\hspace{0.33em}\Rightarrow \hspace{0.33em}\frac{{p}_{n}}{{p}_{1}}\lt \frac{n+2}{n}.We want to conclude this introduction with a brief outline of the proof of Theorem 1.4. The main idea, which was also used in [6,15,22], is to test the weak formulation (1.3) with a finite difference of uu, i.e., φ(x,t)≈u(x+hes,t)−u(x,t)\varphi \left(x,t)\approx u\left(x+h{e}_{s},t)-u\left(x,t). Via a Caccioppoli type inequality, we will then strive to obtain a uniform bound of the type (1.6)∫Qϱ(z0)∣Diu(x+hes,t)−Diu(x,t)∣pi∣h∣γpidz≤C,\mathop{\int }\limits_{{Q}_{\varrho }\left({z}_{0})}\frac{{| {D}_{i}u\left(x+h{e}_{s},t)-{D}_{i}u\left(x,t)| }^{{p}_{i}}}{{| h| }^{\gamma {p}_{i}}}{\rm{d}}z\le C,locally on any cylinder Qϱ(z0){Q}_{\varrho }\left({z}_{0}), where γ∈(0,1)\gamma \in \left(0,1)and Qϱ(z0)=Bϱ(x0)×(t0−ϱ2,t0){Q}_{\varrho }\left({z}_{0})={B}_{\varrho }\left({x}_{0})\times \left({t}_{0}-{\varrho }^{2},{t}_{0})denotes a standard parabolic cylinder with center z0=(x0,t0)∈ΩT{z}_{0}=\left({x}_{0},{t}_{0})\in {\Omega }_{T}and radius ϱ>0\varrho \gt 0. Roughly speaking, the estimate (1.6) asserts that the Lpi{L}^{{p}_{i}}-norm of a fractional difference quotient is uniformly bounded with respect to hh. From this, we can conclude that uubelongs to a certain fractional Sobolev space, and via an embedding theorem, we obtain a small amount of higher integrability for Diu{D}_{i}u. To obtain the full higher integrability, we then need to perform a suitable iteration procedure in the last step.The article is organized in the following way: In Section 2, we gather some preliminaries, which will be needed later on. In Section 3, we prove the Caccioppoli type inequality for finite differences. The iteration procedure, that is needed for the proof of Theorem 1.4, will be performed in the last section.2PreliminariesIn this chapter, we gather some important tools that will be needed for the proof of the higher integrability, in particular, concerning fractional Sobolev spaces and difference quotients respectively finite differences.2.1Some useful inequalitiesTo absorb certain terms, we will need the following standard iteration lemma [21, Lemma 6.1].Lemma 2.1Let ϑ∈(0,1){\vartheta }\in \left(0,1), A,B≥0A,B\ge 0, and α>0\alpha \gt 0. There exists a constant C(α,ϑ)C\left(\alpha ,{\vartheta })such that there holds: For any r∈(0,ϱ)r\in \left(0,\varrho )and any nonnegative, bounded function Φ:[r,ϱ]→[0,∞)\Phi :\left[r,\varrho ]\to \left[0,\infty )satisfyingΦ(s)≤ϑΦ(t)+A(t−s)α+B∀r≤s<t≤ϱ,\Phi \left(s)\le {\vartheta }\Phi \left(t)+\frac{A}{{\left(t-s)}^{\alpha }}+B\hspace{1.0em}\forall r\le s\lt t\le \varrho ,we haveΦ(r)≤CA(ϱ−r)α+B.\Phi \left(r)\le C\left[\frac{A}{{\left(\varrho -r)}^{\alpha }}+B\right].For the proof of the Caccioppoli inequality, we will need the following technical lemma. The case σ<0\sigma \lt 0was proved in [1, Lemma 2.1] and the case σ≥0\sigma \ge 0in [20, Lemma 2.1].Lemma 2.2Let k∈Nk\in {\mathbb{N}}. For every σ>−12\sigma \gt -\frac{1}{2}, there exists a constant C=C(σ)≥1C=C\left(\sigma )\ge 1, such that the following estimate holds: 1C(μ2+∣A∣2+∣B∣2)σ≤∫01(μ2+∣A+s(B−A)∣2)σds≤C(μ2+∣A∣2+∣B∣2)σ\frac{1}{C}{({\mu }^{2}+{| A| }^{2}+{| B| }^{2})}^{\sigma }\le \underset{0}{\overset{1}{\int }}{({\mu }^{2}+{| A+s\left(B-A)| }^{2})}^{\sigma }{\rm{d}}s\le C{({\mu }^{2}+{| A| }^{2}+{| B| }^{2})}^{\sigma }for any μ≥0\mu \ge 0and any A,B∈RkA,B\in {{\mathbb{R}}}^{k}, not both zero if μ=0\mu =0and σ<0\sigma \lt 0.The following lemma [30, Lemma 3.2] is a parabolic version of the Sobolev inequality and follows from the Gagliardo-Nirenberg inequality.Lemma 2.3Let σ≥1\sigma \ge 1, Qϱ(z0)⊂ΩT{Q}_{\varrho }\left({z}_{0})\subset {\Omega }_{T}andu∈Lσ(t0−ϱ2,t0;W1,σ(Bϱ(x0),RN))∩L∞(t0−ϱ2,t0;L2(Bϱ(x0),RN)).u\in {L}^{\sigma }({t}_{0}-{\varrho }^{2},{t}_{0};\hspace{0.33em}{W}^{1,\sigma }({B}_{\varrho }\left({x}_{0}),{{\mathbb{R}}}^{N}))\cap {L}^{\infty }({t}_{0}-{\varrho }^{2},{t}_{0};\hspace{0.33em}{L}^{2}({B}_{\varrho }\left({x}_{0}),{{\mathbb{R}}}^{N})).There exists a constant C=C(N,n,σ)C=C\left(N,n,\sigma ), such that for any radius ϱ2≤r<ϱ\frac{\varrho }{2}\le r\lt \varrho , the following estimate holds: ∫Qr(z0)∣u∣σ(n+2)ndz≤C∫Qϱ(z0)∣Du∣σ+uϱ−rσdzsupt∈(t0−ϱ2,t0)∫Bϱ(x0)∣u(⋅,t)∣2dxσn.\mathop{\int }\limits_{{Q}_{r}\left({z}_{0})}{| u| }^{\tfrac{\sigma \left(n+2)}{n}}{\rm{d}}z\le C\mathop{\int }\limits_{{Q}_{\varrho }\left({z}_{0})}\left({| Du| }^{\sigma }+{\left|\frac{u}{\varrho -r}\right|}^{\sigma }\right){\rm{d}}z{\left(\mathop{\sup }\limits_{t\in \left({t}_{0}-{\varrho }^{2},{t}_{0})}\mathop{\int }\limits_{{B}_{\varrho }\left({x}_{0})}{| u\left(\cdot ,t)| }^{2}{\rm{d}}x\right)}^{\tfrac{\sigma }{n}}.2.2Finite differences and fractional Sobolev spacesLet v:ΩT→RNv:{\Omega }_{T}\to {{\mathbb{R}}}^{N}be some function. By τs,hv:ΩTh→RN{\tau }_{s,h}v:{\Omega }_{T}^{h}\to {{\mathbb{R}}}^{N}, we denote the finite difference of vvin the spatial direction s∈{1,…,n}s\in \{1,\ldots ,n\}with increment h∈Rh\in {\mathbb{R}}, i.e., τs,hv(x,t)=v(x+hes,t)−v(x,t).{\tau }_{s,h}v\left(x,t)=v\left(x+h{e}_{s},t)-v\left(x,t).By ΩTh={x∈Ω:dist(x,∂Ω)>∣h∣}×(0,T){\Omega }_{T}^{h}=\{x\in \Omega :{\rm{dist}}\left(x,\partial \Omega )\gt | h| \}\times \left(0,T), we have denoted the inner parallel cylinder (with respect to space) at distance ∣h∣| h| . We will need the following simple estimate.Lemma 2.4Let Q2R(z0)⊂ΩT{Q}_{2R}\left({z}_{0})\subset {\Omega }_{T}, p∈[1,∞)p\in \left[1,\infty ), and f∈Lp(Q2R(z0))f\in {L}^{p}\left({Q}_{2R}\left({z}_{0})). There exists a positive constant C=C(p)C=C\left(p), such that for any h∈(−R,R)h\in \left(-R,R), the following inequality holds: (2.1)∫QR(1+∣f∣+∣τs,hf∣)pdz≤C∫Q2R(1+∣f∣)pdz.\mathop{\int }\limits_{{Q}_{R}}{\left(1+| f| +| {\tau }_{s,h}f| )}^{p}{\rm{d}}z\le C\mathop{\int }\limits_{{Q}_{2R}}{\left(1+| f| )}^{p}{\rm{d}}z.The next lemma, which can be found, e.g., in [17, Chapter 5.8.2, Theorem 3], asserts that the Lp{L}^{p}-norm of a difference quotient on some cylinder is bounded by the Lp{L}^{p}-norm of the respective partial derivative on a larger cylinder.Lemma 2.5Let f,Dsf∈Lp(Q2R)f,{D}_{s}f\in {L}^{p}\left({Q}_{2R})with s∈{1,…,n}s\in \{1,\ldots ,n\}and p∈[1,∞)p\in \left[1,\infty ). Then, for any h∈(−R,R)h\in \left(-R,R), the following inequality holds: (2.2)∫QR∣τs,hf∣pdz≤∣h∣p∫Q2R∣Dsf∣pdz.\mathop{\int }\limits_{{Q}_{R}}{| {\tau }_{s,h}f| }^{p}{\rm{d}}z\le {| h| }^{p}\mathop{\int }\limits_{{Q}_{2R}}{| {D}_{s}f| }^{p}{\rm{d}}z.Remark 2.6Obviously, the last inequality can be rewritten as follows: ∫QRf(x+hes,t)−f(x,t)hpdz≤∫Q2R∣Dsf∣pdz,\mathop{\int }\limits_{{Q}_{R}}{\left|\frac{f\left(x+h{e}_{s},t)-f\left(x,t)}{h}\right|}^{p}{\rm{d}}z\le \mathop{\int }\limits_{{Q}_{2R}}{| {D}_{s}f| }^{p}{\rm{d}}z,i.e., the Lp{L}^{p}-norm of the difference quotient in direction ss(on a parabolic cylinder) is bounded by the Lp{L}^{p}-norm of the partial derivative Dsf{D}_{s}fon a larger cylinder, and this bound is uniform with respect to hh.Later on, we will use these finite differences to show that the partial derivatives of a weak solution belong to some fractional Sobolev space, from which the higher integrability follows via an embedding theorem. Thus, we also need a few results about elliptic and parabolic fractional Sobolev spaces. Let us first recall the definition of a fractional Sobolev space in the elliptic setting [11, Section 2]. Let k∈N0k\in {{\mathbb{N}}}_{0}and p≥1p\ge 1. We say that a function f∈Wk,p(Ω,RN)f\in {W}^{k,p}(\Omega ,{{\mathbb{R}}}^{N})belongs to the fractional Sobolev space (or Sobolev-Slobodeckij space) Wk+α,p(Ω,RN){W}^{k+\alpha ,p}(\Omega ,{{\mathbb{R}}}^{N}), for some α∈(0,1)\alpha \in \left(0,1), if the Gagliardo semi-norm [Dβf]α,p;Ωp≔∫Ω∫Ω∣Dβf(x)−Dβf(y)∣p∣x−y∣n+αpdxdy{[}{D}^{\beta }f{]}_{\alpha ,p;\hspace{0.33em}\Omega }^{p}:= \mathop{\int }\limits_{\Omega }\mathop{\int }\limits_{\Omega }\frac{| {D}^{\beta }f\left(x)-{D}^{\beta }f(y){| }^{p}}{{| x-y| }^{n+\alpha p}}{\rm{d}}x{\rm{d}}yis finite for any multiindex β∈N0n\beta \in {{\mathbb{N}}}_{0}^{n}with ∣β∣=k| \beta | =k. The space Wk+α,p(Ω,RN){W}^{k+\alpha ,p}(\Omega ,{{\mathbb{R}}}^{N}), endowed with the norm ‖f‖Wk+α,p(Ω)≔‖f‖Wk,p(Ω)+∑∣β∣=k[Dβf]α,p;Ω,{\Vert f\Vert }_{{W}^{k+\alpha ,p}\left(\Omega )}:= {\Vert f\Vert }_{{W}^{k,p}\left(\Omega )}+\sum _{| \beta | =k}{[}{D}^{\beta }f{]}_{\alpha ,p;\Omega },is a Banach space. For later use, we state the following interpolation result for fractional Sobolev spaces, which is essentially an anisotropic version of the interpolation result from [7, Corollary 3.2]. The proof can be found in the appendix.Lemma 2.7Let λ,μ∈(0,1)\lambda ,\mu \in \left(0,1)and p∈(1,∞)p\in \left(1,\infty ). Let α∈(0,1)\alpha \in \left(0,1)be such that1+α=θ(1+λ)+(1−θ)μ1+\alpha =\theta \left(1+\lambda )+\left(1-\theta )\mu for some θ∈(0,1)\theta \in \left(0,1). Furthermore, let1r=θp+1−θ2.\frac{1}{r}=\frac{\theta }{p}+\frac{1-\theta }{2}.Finally, let f∈W1,p(Rn)∩Wμ,2(Rn)f\in {W}^{1,p}\left({{\mathbb{R}}}^{n})\cap {W}^{\mu ,2}\left({{\mathbb{R}}}^{n})and D1f∈Wλ,p(Rn){D}_{1}f\in {W}^{\lambda ,p}\left({{\mathbb{R}}}^{n}). Then we have D1f∈Wα,r(Rn){D}_{1}f\in {W}^{\alpha ,r}\left({{\mathbb{R}}}^{n}), and the estimate(2.3)‖D1f‖Wα,r≤C[‖D1f‖Lp+[D1f]λ,p]θ‖f‖Wμ,21−θ{\Vert {D}_{1}f\Vert }_{{W}^{\alpha ,r}}\le C{{[}{\Vert {D}_{1}f\Vert }_{{L}^{p}}+{\left[{D}_{1}f]}_{\lambda ,p}]}^{\theta }{\Vert f\Vert }_{{W}^{\mu ,2}}^{1-\theta }holds.Remark 2.8The estimate (2.3) also holds for all α∈(0,1)\alpha \in \left(0,1)such that 1+α≤θ(1+λ)+(1−θ)μ.1+\alpha \le \theta \left(1+\lambda )+\left(1-\theta )\mu .This follows from the continuous embedding Wα,r(Rn)↪Wα′,r(Rn){W}^{\alpha ,r}\left({{\mathbb{R}}}^{n})\hspace{0.33em}\hookrightarrow \hspace{0.33em}{W}^{\alpha ^{\prime} ,r}\left({{\mathbb{R}}}^{n})for 0<α′≤α<10\lt \alpha ^{\prime} \le \alpha \lt 1, see, e.g., [11, Proposition 2.1].We also recall the following classical embedding theorem, which can be found in [11, Theorem 6.5].Lemma 2.9Let α∈(0,1)\alpha \in \left(0,1)and p∈[1,∞)p\in \left[1,\infty )such that αp<n\alpha p\lt n. There exists a constant C(n,p,α)C\left(n,p,\alpha )such that for any f∈Wα,p(Rn)f\in {W}^{\alpha ,p}\left({{\mathbb{R}}}^{n}), we have‖f‖Lnpn−αp(Rn)≤C[f]α,p;Rn.{\Vert f\Vert }_{{L}^{\tfrac{np}{n-\alpha p}}\left({{\mathbb{R}}}^{n})}\le C\hspace{0.33em}{[f]}_{\alpha ,p;{{\mathbb{R}}}^{n}}.Hence, the space Wα,p(Rn){W}^{\alpha ,p}\left({{\mathbb{R}}}^{n})is continuously embedded into Lnpn−αp(Rn){L}^{\tfrac{np}{n-\alpha p}}\left({{\mathbb{R}}}^{n}).With these prerequisites at hand, we can prove the following fractional version of the Gagliardo-Nirenberg inequality.Lemma 2.10Let Bϱ(x0)⊂Rn{B}_{\varrho }\left({x}_{0})\subset {{\mathbb{R}}}^{n}be a ball and let λ,μ,θ∈(0,1)\lambda ,\mu ,\theta \in \left(0,1), 1<p,2<s<∞1\lt p,2\lt s\lt \infty , such that(2.4)−ns≤θλ−np−(1−θ)1−μ+n2.-\frac{n}{s}\le \theta \left(\lambda -\frac{n}{p}\right)-\left(1-\theta )\left(1-\mu +\frac{n}{2}\right).Suppose that f∈W1,p(Bϱ(x0))∩Wμ,2(Bϱ(x0))f\in {W}^{1,p}\left({B}_{\varrho }\left({x}_{0}))\cap {W}^{\mu ,2}\left({B}_{\varrho }\left({x}_{0}))and Dif∈Wλ,p(Bϱ(x0)){D}_{i}f\in {W}^{\lambda ,p}\left({B}_{\varrho }\left({x}_{0}))for a fixed index i∈{1,…,n}i\in \{1,\ldots ,n\}. Then Dif∈Ls(Br(x0)){D}_{i}f\in {L}^{s}\left({B}_{r}\left({x}_{0}))for any radius r∈(0,ϱ)r\in \left(0,\varrho ), and there exists a constant C(n,μ,λ,p,s,θ,1/(ϱ−r))C\left(n,\mu ,\lambda ,p,s,\theta ,1\hspace{0.1em}\text{/}\hspace{0.1em}\left(\varrho -r))such that‖Dif‖Ls(Br(x0))≤C(‖f‖W1,p(Bϱ(x0))+[Dif]λ,p;Bϱ(x0))θ‖f‖Wμ,2(Bϱ(x0))1−θ.{\Vert {D}_{i}f\Vert }_{{L}^{s}\left({B}_{r}\left({x}_{0}))}\le C{({\Vert f\Vert }_{{W}^{1,p}\left({B}_{\varrho }\left({x}_{0}))}+{{[}{D}_{i}f]}_{\lambda ,p;{B}_{\varrho }\left({x}_{0})})}^{\theta }{\Vert f\Vert }_{{W}^{\mu ,2}\left({B}_{\varrho }\left({x}_{0}))}^{1-\theta }.The aforementioned lemma is a slightly modified version of [6, Lemma 6.4]. The difference lies in the fact that we do not obtain the higher integrability of DfDf, but only of a fixed partial derivative Dif{D}_{i}f. To obtain the higher integrability of Dif{D}_{i}f, it is sufficient to presuppose that Dif∈Wλ,p{D}_{i}f\in {W}^{\lambda ,p}, which is a weaker condition than f∈W1+λ,pf\in {W}^{1+\lambda ,p}. As we will see later, this is a necessary adaptation to the anisotropic setting. The proof is very similar to [6], but we include it for completeness.ProofTo simplify the notation, we suppress the center of the ball and write Bϱ{B}_{\varrho }instead of Bϱ(x0){B}_{\varrho }\left({x}_{0}). Let η∈C0∞(Bϱ)\eta \in {C}_{0}^{\infty }\left({B}_{\varrho })be a cutoff function such that 0≤η≤10\le \eta \le 1, η=1\eta =1on Br{B}_{r}and ∣Dη∣≤2ϱ−r| D\eta | \le \frac{2}{\varrho -r}. Now we choose the parameter α\alpha such that (2.5)n+αsns=θp+1−θ2⇔α=θnp+(1−θ)n2−ns.\frac{n+\alpha s}{ns}=\frac{\theta }{p}+\frac{1-\theta }{2}\hspace{0.33em}\iff \hspace{0.33em}\alpha =\frac{\theta n}{p}+\frac{\left(1-\theta )n}{2}-\frac{n}{s}.Due to (2.4), we have (2.6)α≤θnp+(1−θ)n2+θλ−np−(1−θ)1−μ+n2=θλ−(1−θ)(1−μ)<1.\alpha \le \frac{\theta n}{p}+\frac{\left(1-\theta )n}{2}+\theta \left(\lambda -\frac{n}{p}\right)-\left(1-\theta )\left(1-\mu +\frac{n}{2}\right)=\theta \lambda -\left(1-\theta )\left(1-\mu )\lt 1.Since p,2<sp,2\lt s, we also have α=θnp−ns+(1−θ)n2−ns>0,\alpha =\theta \left(\frac{n}{p}-\frac{n}{s}\right)+\left(1-\theta )\left(\frac{n}{2}-\frac{n}{s}\right)\gt 0,and hence, α∈(0,1)\alpha \in \left(0,1). Therefore, we can apply Lemma 2.9 with α,nsn+αs\left(\alpha ,\frac{ns}{n+\alpha s}\right)instead of (α,p)\left(\alpha ,p)to infer that ‖Dif‖Ls(Br)≤‖Di(fη)‖Ls(Rn)≤C(n,s,α)[Di(fη)]α,nsn+αs;Rn.{\Vert {D}_{i}f\Vert }_{{L}^{s}\left({B}_{r})}\le {\Vert {D}_{i}(f\eta )\Vert }_{{L}^{s}\left({{\mathbb{R}}}^{n})}\le C\left(n,s,\alpha ){{[}{D}_{i}(f\eta )]}_{\alpha ,\tfrac{ns}{n+\alpha s};{{\mathbb{R}}}^{n}}.Due to the upper bound (2.6), which is a consequence of the choice of α\alpha from (2.5), we have 1+α≤θ(1+λ)+(1−θ)μ1+\alpha \le \theta \left(1+\lambda )+\left(1-\theta )\mu . Hence, we can apply Lemma 2.7 resp. Remark 2.8 with nsn+αs\frac{ns}{n+\alpha s}instead of rrto the right-hand side. This yields [Di(fη)]α,nsn+αs;Rn≤C(‖Di(fη)‖Lp(Rn)+[Di(fη)]λ,p;Rn)θ‖fη‖Wμ,2(Rn)1−θ≤C(‖Di(fη)‖Lp(Bϱ)+[Di(fη)]λ,p;Bϱ)θ‖f‖Wμ,2(Bϱ)1−θ.\begin{array}{rcl}{{[}{D}_{i}(f\eta )]}_{\alpha ,\tfrac{ns}{n+\alpha s};{{\mathbb{R}}}^{n}}& \le & C{({\Vert {D}_{i}(f\eta )\Vert }_{{L}^{p}\left({{\mathbb{R}}}^{n})}+{{[}{D}_{i}(f\eta )]}_{\lambda ,p;{{\mathbb{R}}}^{n}})}^{\theta }{\Vert f\eta \Vert }_{{W}^{\mu ,2}\left({{\mathbb{R}}}^{n})}^{1-\theta }\\ & \le & C{({\Vert {D}_{i}(f\eta )\Vert }_{{L}^{p}\left({B}_{\varrho })}+{{[}{D}_{i}(f\eta )]}_{\lambda ,p;{B}_{\varrho }})}^{\theta }{\Vert f\Vert }_{{W}^{\mu ,2}\left({B}_{\varrho })}^{1-\theta }.\end{array}In this estimate, the constant CCdepends on λ,μ,p\lambda ,\mu ,p, and θ\theta . Now we use the fact that ∣η∣≤1,∣Dη∣≤2ϱ−r| \eta | \le 1,| D\eta | \le \frac{2}{\varrho -r}to estimate the Gagliardo-Seminorm on the right-hand side in the following way: [Di(fη)]λ,p;Bϱ=∫Bϱ∫Bϱ∣Dif(x)η(x)−Dif(y)η(y)+f(x)Diη(x)−f(y)Diη(y)∣p∣x−y∣n+λpdxdy1p≤Cp,1ϱ−r[Dif]λ,p;Bϱ+∫Bϱ∫Bϱ∣f(x)−f(y)∣p∣x−y∣n+λpdxdy1p=Cp,1ϱ−r[[Dif]λ,p;Bϱ+[f]λ,p;Bϱ]≤Cp,1ϱ−r[‖f‖W1,p(Bϱ)+[Dif]λ,p;Bϱ].\begin{array}{rcl}{\left[{D}_{i}(f\eta )]}_{\lambda ,p;{B}_{\varrho }}& =& {\left(\mathop{\displaystyle \int }\limits_{{B}_{\varrho }}\mathop{\displaystyle \int }\limits_{{B}_{\varrho }}\frac{{| {D}_{i}f\left(x)\eta \left(x)-{D}_{i}f(y)\eta (y)+f\left(x){D}_{i}\eta \left(x)-f(y){D}_{i}\eta (y)| }^{p}}{{| x-y| }^{n+\lambda p}}{\rm{d}}x{\rm{d}}y\right)}^{\tfrac{1}{p}}\\ & \le & C\left(p,\frac{1}{\varrho -r}\right)\left[{{[}{D}_{i}f]}_{\lambda ,p;{B}_{\varrho }}+{\left(\mathop{\displaystyle \int }\limits_{{B}_{\varrho }}\mathop{\displaystyle \int }\limits_{{B}_{\varrho }}\frac{{| f\left(x)-f(y)| }^{p}}{{| x-y| }^{n+\lambda p}}{\rm{d}}x{\rm{d}}y\right)}^{\tfrac{1}{p}}\right]\\ & =& C\left(p,\frac{1}{\varrho -r}\right){[}{{[}{D}_{i}f]}_{\lambda ,p;{B}_{\varrho }}+{{[}f]}_{\lambda ,p;{B}_{\varrho }}]\\ & \le & C\left(p,\frac{1}{\varrho -r}\right){[}{\Vert f\Vert }_{{W}^{1,p}\left({B}_{\varrho })}+{{[}{D}_{i}f]}_{\lambda ,p;{B}_{\varrho }}].\end{array}Similarly, we also have ‖Di(fη)‖Lp(Bϱ)≤Cp,1ϱ−r‖f‖W1,p(Bϱ){\Vert {D}_{i}(f\eta )\Vert }_{{L}^{p}\left({B}_{\varrho })}\le C\left(p,\frac{1}{\varrho -r}\right){\Vert f\Vert }_{{W}^{1,p}\left({B}_{\varrho })}. Hence, we obtain the desired estimate ‖Dif‖Ls(Br)≤C(‖f‖W1,p(Bϱ)+[Dif]λ,p;Bϱ)θ‖f‖Wμ,2(Bϱ)1−θ,{\Vert {D}_{i}f\Vert }_{{L}^{s}\left({B}_{r})}\le C{({\Vert f\Vert }_{{W}^{1,p}\left({B}_{\varrho })}+{{[}{D}_{i}f]}_{\lambda ,p;{B}_{\varrho }})}^{\theta }{\Vert f\Vert }_{{W}^{\mu ,2}\left({B}_{\varrho })}^{1-\theta },with a constant that depends on n,s,α,λ,μ,p,θn,s,\alpha ,\lambda ,\mu ,p,\theta , and 1ϱ−r\frac{1}{\varrho -r}. Since α\alpha itself depends on θ,n,p\theta ,n,p, and ss(see (2.5)), we ultimately have C=C(n,μ,λ,p,s,θ,1/(ϱ−r))C=C\left(n,\mu ,\lambda ,p,s,\theta ,1\hspace{0.1em}\text{/}\hspace{0.1em}\left(\varrho -r)).□Of course, we will also need a parabolic version of such fractional Sobolev spaces. Let k∈N0k\in {{\mathbb{N}}}_{0}, p≥1p\ge 1and α∈(0,1)\alpha \in \left(0,1). A function u∈Lp(0,T;Wk,p(Ω,RN))u\in {L}^{p}(0,T;\hspace{0.33em}{W}^{k,p}(\Omega ,{{\mathbb{R}}}^{N}))belongs to the parabolic fractional Sobolev space Lp(0,T;Wk+α,p(Ω,RN)){L}^{p}(0,T;\hspace{0.33em}{W}^{k+\alpha ,p}(\Omega ,{{\mathbb{R}}}^{N})), if the parabolic Gagliardo semi-norm [Dβu]α,0,p;ΩTp≔∫0T∫Ω∫Ω∣Dβu(x,t)−Dβu(y,t)∣p∣x−y∣n+αpdxdydt{[}{D}^{\beta }u{]}_{\alpha ,0,p;\hspace{0.33em}{\Omega }_{T}}^{p}:= \underset{0}{\overset{T}{\int }}\mathop{\int }\limits_{\Omega }\mathop{\int }\limits_{\Omega }\frac{| {D}^{\beta }u\left(x,t)-{D}^{\beta }u(y,t){| }^{p}}{{| x-y| }^{n+\alpha p}}{\rm{d}}x{\rm{d}}y{\rm{d}}tis finite for any β∈N0n\beta \in {{\mathbb{N}}}_{0}^{n}with ∣β∣=k| \beta | =k. Similarly as in the elliptic case, the space Lp(0,T;Wk+α,p(Ω,RN)){L}^{p}(0,T;\hspace{0.33em}{W}^{k+\alpha ,p}(\Omega ,{{\mathbb{R}}}^{N})), endowed with the norm ‖u‖k+α,0,p;ΩT≔‖u‖Lp(0,T;Wk,p(Ω,RN))+∑∣β∣=k[Dβf]α,0,p;ΩT,{\Vert u\Vert }_{k+\alpha ,0,p;{\Omega }_{T}}:= {\Vert u\Vert }_{{L}^{p}\left(0,T;{W}^{k,p}\left(\Omega ,{{\mathbb{R}}}^{N}))}+\sum _{| \beta | =k}{[}{D}^{\beta }f{]}_{\alpha ,0,p;{\Omega }_{T}},is a Banach space. The following lemma is an anisotropic version of the parabolic fractional Sobolev inequality from [6, Lemma 6.5].Lemma 2.11Let Qϱ(z0)⊂ΩT{Q}_{\varrho }\left({z}_{0})\subset {\Omega }_{T}be a parabolic cylinder with radius ϱ≤1\varrho \le 1. Let λ,μ∈(0,1)\lambda ,\mu \in \left(0,1), 1<p,2<s<∞1\lt p,2\lt s\lt \infty be parameters such that(2.7)(s−p)1−μ+n2≤λp.\left(s-p)\left(1-\mu +\frac{n}{2}\right)\le \lambda p.Furthermore, let us assume thatu∈Lp(t0−ϱ2,t0;W1,p(Bϱ(x0)))∩L∞(t0−ϱ2,t0;Wμ,2(Bϱ(x0)))u\in {L}^{p}\left({t}_{0}-{\varrho }^{2},{t}_{0};\hspace{0.33em}{W}^{1,p}\left({B}_{\varrho }\left({x}_{0})))\cap {L}^{\infty }\left({t}_{0}-{\varrho }^{2},{t}_{0};\hspace{0.33em}{W}^{\mu ,2}\left({B}_{\varrho }\left({x}_{0})))andDiu∈Lp(t0−ϱ2,t0;Wλ,p(Bϱ(x0))){D}_{i}u\in {L}^{p}\left({t}_{0}-{\varrho }^{2},{t}_{0};\hspace{0.33em}{W}^{\lambda ,p}\left({B}_{\varrho }\left({x}_{0})))for a fixed index i∈{1,…,n}i\in \{1,\ldots ,n\}. Then we have Diu∈Ls(Br(x0)×(t0−ϱ2,t0)){D}_{i}u\in {L}^{s}\left({B}_{r}\left({x}_{0})\times \left({t}_{0}-{\varrho }^{2},{t}_{0}))for any radius r∈(0,ϱ)r\in \left(0,\varrho ), and there exists a constant C(n,μ,λ,p,s,1/(ϱ−r))C\left(n,\mu ,\lambda ,p,s,1\hspace{0.1em}\text{/}\hspace{0.1em}\left(\varrho -r))such that the following estimate holds: ‖Diu‖Ls(Br(x0)×(t1,t0))≤C(‖u‖Lp(t1,t0;W1,p(Bϱ))+[Diu]λ,0,p;Bϱ×(t1,t0))pssupt∈(t1,t0)‖u(⋅,t)‖Wμ,2(Bϱ)s−ps,\Vert {D}_{i}u{\Vert }_{{L}^{s}\left({B}_{r}\left({x}_{0})\times \left({t}_{1},{t}_{0}))}\le C{({\Vert u\Vert }_{{L}^{p}\left({t}_{1},{t}_{0};{W}^{1,p}\left({B}_{\varrho }))}+{\left[{D}_{i}u]}_{\lambda ,0,p;{B}_{\varrho }\times \left({t}_{1},{t}_{0})})}^{\tfrac{p}{s}}\mathop{\sup }\limits_{t\in \left({t}_{1},{t}_{0})}{\Vert u\left(\cdot ,t)\Vert }_{{W}^{\mu ,2}\left({B}_{\varrho })}^{\frac{s-p}{s}},where we used the abbreviations t1≔t0−ϱ2{t}_{1}:= {t}_{0}-{\varrho }^{2}and Bϱ≔Bϱ(x0){B}_{\varrho }:= {B}_{\varrho }\left({x}_{0}).ProofTo simplify the notation, we suppress the center of the ball and write Bϱ{B}_{\varrho }instead of Bϱ(x0){B}_{\varrho }\left({x}_{0}). For almost every time-slice t∈(t1,t0)t\in \left({t}_{1},{t}_{0}), we have u(⋅,t)∈W1,p(Bϱ)∩Wμ,2(Bϱ)u\left(\cdot ,t)\in {W}^{1,p}\left({B}_{\varrho })\cap {W}^{\mu ,2}\left({B}_{\varrho })and Diu(⋅,t)∈Wλ,p(Bϱ){D}_{i}u\left(\cdot ,t)\in {W}^{\lambda ,p}\left({B}_{\varrho }). Hence, we can apply Lemma 2.10 with θ=ps\theta =\frac{p}{s}. We note that the condition (2.4) is satisfied due to (2.7). We obtain ‖Diu(⋅,t)‖Ls(Br)≤C(‖u(⋅,t)‖W1,p(Bϱ)+[Diu(⋅,t)]λ,p;Bϱ)ps‖u(⋅,t)‖Wμ,2(Bϱ)s−ps{\Vert {D}_{i}u\left(\cdot ,t)\Vert }_{{L}^{s}\left({B}_{r})}\le C{({\Vert u\left(\cdot ,t)\Vert }_{{W}^{1,p}\left({B}_{\varrho })}+{{[}{D}_{i}u\left(\cdot ,t)]}_{\lambda ,p;{B}_{\varrho }})}^{\tfrac{p}{s}}{\Vert u\left(\cdot ,t)\Vert }_{{W}^{\mu ,2}\left({B}_{\varrho })}^{\frac{s-p}{s}}for almost every time-slice t∈(t1,t0)t\in \left({t}_{1},{t}_{0}). We integrate this inequality with respect to time to obtain ∫t1t0∫Br∣Diu∣sdxdt≤C∫t1t0(‖u(⋅,t)‖W1,p(Bϱ)+[Diu(⋅,t)]λ,p;Bϱ)p‖u(⋅,t)‖Wμ,2(Bϱ)s−pdt≤C∫t1t0(‖u(⋅,t)‖W1,p(Bϱ)+[Diu(⋅,t)]λ,p;Bϱ)pdtsupt∈(t1,t0)‖u(⋅,t)‖Wμ,2(Bϱ)s−p≤C(‖u‖Lp(t1,t0;W1,p(Bϱ))+[Diu]λ,0,p;Bϱ×(t1,t0))psupt∈(t1,t0)‖u(⋅,t)‖Wμ,2(Bϱ)s−p.\begin{array}{rcl}\underset{{t}_{1}}{\overset{{t}_{0}}{\displaystyle \int }}\mathop{\displaystyle \int }\limits_{{B}_{r}}{| {D}_{i}u| }^{s}{\rm{d}}x{\rm{d}}t& \le & C\underset{{t}_{1}}{\overset{{t}_{0}}{\displaystyle \int }}{({\Vert u\left(\cdot ,t)\Vert }_{{W}^{1,p}\left({B}_{\varrho })}+{{[}{D}_{i}u\left(\cdot ,t)]}_{\lambda ,p;{B}_{\varrho }})}^{p}{\Vert u\left(\cdot ,t)\Vert }_{{W}^{\mu ,2}\left({B}_{\varrho })}^{s-p}{\rm{d}}t\\ & \le & C\underset{{t}_{1}}{\overset{{t}_{0}}{\displaystyle \int }}{({\Vert u\left(\cdot ,t)\Vert }_{{W}^{1,p}\left({B}_{\varrho })}+{{[}{D}_{i}u\left(\cdot ,t)]}_{\lambda ,p;{B}_{\varrho }})}^{p}{\rm{d}}t\mathop{\sup }\limits_{t\in \left({t}_{1},{t}_{0})}{\Vert u\left(\cdot ,t)\Vert }_{{W}^{\mu ,2}\left({B}_{\varrho })}^{s-p}\\ & \le & C{({\Vert u\Vert }_{{L}^{p}\left({t}_{1},{t}_{0};{W}^{1,p}\left({B}_{\varrho }))}+{{[}{D}_{i}u]}_{\lambda ,0,p;{B}_{\varrho }\times \left({t}_{1},{t}_{0})})}^{p}\mathop{\sup }\limits_{t\in \left({t}_{1},{t}_{0})}{\Vert u\left(\cdot ,t)\Vert }_{{W}^{\mu ,2}\left({B}_{\varrho })}^{s-p}.\end{array}In these calculations, the constant CCdepends on n,μ,λ,p,s,θ,1ϱ−rn,\mu ,\lambda ,p,s,\theta ,\frac{1}{\varrho -r}. Since θ=θ(p,s)\theta =\theta \left(p,s), we ultimately have C=C(n,μ,λ,p,s,1/(ϱ−r))C=C\left(n,\mu ,\lambda ,p,s,1\hspace{0.1em}\text{/}\hspace{0.1em}\left(\varrho -r)). This proves the assertion of the lemma.□We will also need the following embedding results for parabolic Nikolskii spaces, which are defined via finite differences. The first part follows from [3, 7.73], the second part is taken from [12, Proposition 2.19]. Roughly speaking, the lemma asserts that uubelongs to certain fractional Sobolev spaces if certain integral norms of a “fractional difference quotient” are uniformly bounded with respect to hh.Lemma 2.12Let Qϱ(z0)⋐ΩT{Q}_{\varrho }\left({z}_{0})\hspace{0.33em}\Subset \hspace{0.33em}{\Omega }_{T}and θ∈(0,1)\theta \in \left(0,1). (1)Assume that u∈L∞(0,T;L2(Ω,RN))u\in {L}^{\infty }(0,T;\hspace{0.33em}{L}^{2}(\Omega ,{{\mathbb{R}}}^{N}))satisfiessupt∈(t0−ϱ2,t0)∫Bϱ(x0)∣u(x+hei,t)−u(x,t)∣2dx≤M∣h∣2θ\mathop{\sup }\limits_{t\in \left({t}_{0}-{\varrho }^{2},{t}_{0})}\mathop{\int }\limits_{{B}_{\varrho }\left({x}_{0})}{| u\left(x+h{e}_{i},t)-u\left(x,t)| }^{2}{\rm{d}}x\le M{| h| }^{2\theta }for every i∈{1,…,n}i\in \{1,\ldots ,n\}and every h∈Rh\in {\mathbb{R}}with ∣h∣<dist(Bϱ(x0),∂Ω)| h| \lt {\rm{dist}}\left({B}_{\varrho }\left({x}_{0}),\partial \Omega ), where M>0M\gt 0is some constant. Then, for every α∈(0,θ)\alpha \in \left(0,\theta )and O⋐Bϱ(x0){\mathcal{O}}\hspace{0.33em}\Subset \hspace{0.33em}{B}_{\varrho }\left({x}_{0}), there exists a constant C(n,θ,α,dist(O,∂Bϱ),dist(Bϱ,∂Ω))C\left(n,\theta ,\alpha ,{\rm{dist}}\left({\mathcal{O}},\partial {B}_{\varrho }),{\rm{dist}}\left({B}_{\varrho },\partial \Omega ))such thatsupt∈(t0−ϱ2,t0)[u(⋅,t)]α,p;O2=supt∈(t0−ϱ2,t0)∫O∫O∣u(x,t)−u(y,t)∣2∣x−y∣n+2αdxdy≤CM.\mathop{\sup }\limits_{t\in \left({t}_{0}-{\varrho }^{2},{t}_{0})}{{[}u\left(\cdot ,t)]}_{\alpha ,p;\hspace{0.33em}{\mathcal{O}}}^{2}=\mathop{\sup }\limits_{t\in \left({t}_{0}-{\varrho }^{2},{t}_{0})}\mathop{\int }\limits_{{\mathcal{O}}}\mathop{\int }\limits_{{\mathcal{O}}}\frac{{| u\left(x,t)-u(y,t)| }^{2}}{{| x-y| }^{n+2\alpha }}{\rm{d}}x{\rm{d}}y\le CM.(2)Assume that u∈Lp(ΩT,RN)u\in {L}^{p}({\Omega }_{T},{{\mathbb{R}}}^{N})satisfies∫t0−ϱ2t0∫Bϱ(x0)∣u(x+hei,t)−u(x,t)∣pdxdt≤M∣h∣θp\underset{{t}_{0}-{\varrho }^{2}}{\overset{{t}_{0}}{\int }}\mathop{\int }\limits_{{B}_{\varrho }\left({x}_{0})}{| u\left(x+h{e}_{i},t)-u\left(x,t)| }^{p}{\rm{d}}x{\rm{d}}t\le M{| h| }^{\theta p}for every i∈{1,…,n}i\in \{1,\ldots ,n\}and every h∈Rh\in {\mathbb{R}}with ∣h∣<dist(Bϱ(x0),∂Ω)| h| \lt {\rm{dist}}\left({B}_{\varrho }\left({x}_{0}),\partial \Omega ), where M>0M\gt 0is some constant. Then, for every γ∈(0,θ)\gamma \in \left(0,\theta )and O⋐Bϱ(x0){\mathcal{O}}\hspace{0.33em}\Subset \hspace{0.33em}{B}_{\varrho }\left({x}_{0}), there exists a constant C(n,θ,γ,dist(O,∂Bϱ),dist(Bϱ,∂Ω))C\left(n,\theta ,\gamma ,{\rm{dist}}\left({\mathcal{O}},\partial {B}_{\varrho }),{\rm{dist}}\left({B}_{\varrho },\partial \Omega ))such that[u]γ,0,p;O×(t0−ϱ2,t0)p=∫t0−ϱ2t0∫O∫O∣u(x,t)−u(y,t)∣p∣x−y∣n+γpdxdydt≤CM.{\left[u]}_{\gamma ,0,p;\hspace{0.33em}{\mathcal{O}}\times \left({t}_{0}-{\varrho }^{2},{t}_{0})}^{p}=\underset{{t}_{0}-{\varrho }^{2}}{\overset{{t}_{0}}{\int }}\mathop{\int }\limits_{{\mathcal{O}}}\mathop{\int }\limits_{{\mathcal{O}}}\frac{{| u\left(x,t)-u(y,t)| }^{p}}{{| x-y| }^{n+\gamma p}}{\rm{d}}x{\rm{d}}y{\rm{d}}t\le CM.3Caccioppoli inequality for finite differencesIn this section, we prove a Caccioppoli type inequality for finite differences, which will be the starting point for the improvement of integrability. The precise formulation reads as follows:Lemma 3.1Let u be a weak solution of (1.1). There exists a constant C(L,ν,pi)C\left(L,\nu ,{p}_{i})such that for any parabolic cylinder Qϱ(z0)⊂ΩT{Q}_{\varrho }\left({z}_{0})\subset {\Omega }_{T}, any radius r∈(0,ϱ)r\in \left(0,\varrho ), any hhwith ∣h∣<dist(Bϱ(x0),∂Ω)| h| \lt {\rm{dist}}\left({B}_{\varrho }\left({x}_{0}),\partial \Omega ), and any direction s∈{1,…,n}s\in \{1,\ldots ,n\}, the following inequality holds: (3.1)supϑ∈(t0−r2,t0)∫Br(x0)∣τs,hu(⋅,ϑ)∣2dx+∑i=1n∫Qr(z0)∣τs,hDiu∣pidz≤C(ϱ−r)2∑i=1n∫Qϱ(z0)(1+∣Diu∣+∣τs,hDiu∣)pi−2∣τs,hu∣2dz.\begin{array}{l}\mathop{\sup }\limits_{{\vartheta }\in \left({t}_{0}-{r}^{2},{t}_{0})}\mathop{\displaystyle \int }\limits_{{B}_{r}\left({x}_{0})}{| {\tau }_{s,h}u\left(\cdot ,{\vartheta })| }^{2}{\rm{d}}x+\mathop{\displaystyle \sum }\limits_{i=1}^{n}\mathop{\displaystyle \int }\limits_{{Q}_{r}\left({z}_{0})}{| {\tau }_{s,h}{D}_{i}u| }^{{p}_{i}}{\rm{d}}z\\ \hspace{1.0em}\le \frac{C}{{\left(\varrho -r)}^{2}}\mathop{\displaystyle \sum }\limits_{i=1}^{n}\mathop{\displaystyle \int }\limits_{{Q}_{\varrho }\left({z}_{0})}{\left(1+| {D}_{i}u| +| {\tau }_{s,h}{D}_{i}u| )}^{{p}_{i}-2}{| {\tau }_{s,h}u| }^{2}{\rm{d}}z.\end{array}ProofTo simplify the notation, we will w.l.o.g. assume that z0=(x0,t0)=(0,0){z}_{0}=\left({x}_{0},{t}_{0})=\left(0,0)and write Qϱ,Bϱ{Q}_{\varrho },{B}_{\varrho }instead of Qϱ(0),Bϱ(0){Q}_{\varrho }\left(0),{B}_{\varrho }\left(0). Let φ∈C0∞(Qϱ,RN)\varphi \in {C}_{0}^{\infty }({Q}_{\varrho },{{\mathbb{R}}}^{N})and ∣h∣<dist(Bϱ,∂Ω)| h| \lt {\rm{dist}}\left({B}_{\varrho },\partial \Omega ). Then we have τs,−hφ∈C0∞(ΩT,RN){\tau }_{s,-h}\varphi \in {C}_{0}^{\infty }({\Omega }_{T},{{\mathbb{R}}}^{N}). By inserting τs,−hφ{\tau }_{s,-h}\varphi into the weak formulation (1.3) and carrying out an “integration by parts for finite differences,” we obtain (3.2)0=∫ΩTu⋅∂t(τs,−hφ)−⟨Df(Du),D(τs,−hφ)⟩dz=∫Qϱτs,hu⋅∂tφ−⟨τs,hDf(Du),Dφ⟩dz.0=\mathop{\int }\limits_{{\Omega }_{T}}u\cdot {\partial }_{t}\left({\tau }_{s,-h}\varphi )-\langle Df\left(Du),D\left({\tau }_{s,-h}\varphi )\rangle {\rm{d}}z=\mathop{\int }\limits_{{Q}_{\varrho }}{\tau }_{s,h}u\cdot {\partial }_{t}\varphi -\langle {\tau }_{s,h}Df\left(Du),D\varphi \rangle {\rm{d}}z.Now we want to insert suitable test functions into (3.2). With respect to space, we choose a cutoff function η∈C0∞(Bϱ)\eta \in {C}_{0}^{\infty }\left({B}_{\varrho })such that 0≤η≤10\le \eta \le 1, η=1\eta =1in Br{B}_{r}and (3.3)∣Dη∣≤2ϱ−r.| D\eta | \le \frac{2}{\varrho -r}.With respect to time, we define a cut-off function ζε∈W01,∞(−ϱ2,0){\zeta }_{\varepsilon }\in {W}_{0}^{1,\infty }(-{\varrho }^{2},0)via (3.4)ζε(t)=1ϱ2−r2(t+ϱ2),on(−ϱ2,−r2),1,on(−r2,ϑ),1ε(ϑ+ε−t),on(ϑ,ϑ+ε),0,on(ϑ+ε,0),{\zeta }_{\varepsilon }\left(t)=\left\{\begin{array}{ll}\frac{1}{{\varrho }^{2}-{r}^{2}}(t+{\varrho }^{2}),\hspace{1.0em}& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}(-{\varrho }^{2},-{r}^{2}),\\ 1,\hspace{1.0em}& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\left(-{r}^{2},{\vartheta }),\\ \frac{1}{\varepsilon }({\vartheta }+\varepsilon -t),\hspace{1.0em}& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\left({\vartheta },{\vartheta }+\varepsilon ),\\ 0,\hspace{1.0em}& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\left({\vartheta }+\varepsilon ,0),\end{array}\right.for some ϑ∈(−r2,0){\vartheta }\in \left(-{r}^{2},0)and ε∈(0,∣ϑ∣)\varepsilon \in \left(0,| {\vartheta }| ). As a test function in the modified weak formulation (3.2), we choose φε(x,t)≔τs,hu(x,t)η2(x)ζε(t).{\varphi }_{\varepsilon }\left(x,t):= {\tau }_{s,h}u\left(x,t){\eta }^{2}\left(x){\zeta }_{\varepsilon }\left(t).Note that this is actually not an admissible test function since it is not smooth and does not even possess a weak derivative with respect to time. The following formal computations, however, can be made rigorous with a standard smoothing procedure with respect to time, as, for instance, via Steklov averages. By inserting φε{\varphi }_{\varepsilon }into (3.2), we obtain −∫Qϱτs,hu⋅∂t(τs,huη2ζε)dz+∫Qϱ⟨τs,hDf(Du),τs,hDu⟩η2ζεdz=−2∫Qϱ⟨τs,hDf(Du),τs,hu⊗Dη⟩ηζεdz.-\mathop{\int }\limits_{{Q}_{\varrho }}{\tau }_{s,h}u\cdot {\partial }_{t}({\tau }_{s,h}u\hspace{0.33em}{\eta }^{2}{\zeta }_{\varepsilon }){\rm{d}}z+\mathop{\int }\limits_{{Q}_{\varrho }}\langle {\tau }_{s,h}Df\left(Du),{\tau }_{s,h}Du\rangle {\eta }^{2}{\zeta }_{\varepsilon }{\rm{d}}z=-2\mathop{\int }\limits_{{Q}_{\varrho }}\langle {\tau }_{s,h}Df\left(Du),{\tau }_{s,h}u\otimes D\eta \rangle \eta {\zeta }_{\varepsilon }{\rm{d}}z.For the first term on the left-hand side, we calculate −∫Qϱτs,hu⋅∂t(τs,huη2ζε)dz=−∫Qϱ∣τs,hu∣2∂tζεη2dz−∫Qϱτs,hu⋅∂t(τs,hu)η2ζεdz=−∫Qϱ∣τs,hu∣2∂tζεη2dz−12∫Qϱ∂t(∣τs,hu∣2)η2ζεdz=−12∫Qϱ∣τs,hu∣2∂tζεη2dz=−12(ϱ2−r2)∫−ϱ2−r2∫Bϱ∣τs,hu∣2η2dxdt+12ε∫ϑϑ+ε∫Bϱ∣τs,hu∣2η2dxdt.\begin{array}{rcl}-\mathop{\displaystyle \int }\limits_{{Q}_{\varrho }}{\tau }_{s,h}u\cdot {\partial }_{t}({\tau }_{s,h}u{\eta }^{2}{\zeta }_{\varepsilon }){\rm{d}}z& =& -\mathop{\displaystyle \int }\limits_{{Q}_{\varrho }}{| {\tau }_{s,h}u| }^{2}{\partial }_{t}{\zeta }_{\varepsilon }{\eta }^{2}{\rm{d}}z-\mathop{\displaystyle \int }\limits_{{Q}_{\varrho }}{\tau }_{s,h}u\cdot {\partial }_{t}\left({\tau }_{s,h}u){\eta }^{2}{\zeta }_{\varepsilon }{\rm{d}}z\\ & =& -\mathop{\displaystyle \int }\limits_{{Q}_{\varrho }}{| {\tau }_{s,h}u| }^{2}{\partial }_{t}{\zeta }_{\varepsilon }{\eta }^{2}{\rm{d}}z-\frac{1}{2}\mathop{\displaystyle \int }\limits_{{Q}_{\varrho }}{\partial }_{t}({| {\tau }_{s,h}u| }^{2}){\eta }^{2}{\zeta }_{\varepsilon }{\rm{d}}z\\ & =& -\frac{1}{2}\mathop{\displaystyle \int }\limits_{{Q}_{\varrho }}{| {\tau }_{s,h}u| }^{2}{\partial }_{t}{\zeta }_{\varepsilon }{\eta }^{2}{\rm{d}}z\\ & =& -\frac{1}{2({\varrho }^{2}-{r}^{2})}\underset{-{\varrho }^{2}}{\overset{-{r}^{2}}{\displaystyle \int }}\mathop{\displaystyle \int }\limits_{{B}_{\varrho }}{| {\tau }_{s,h}u| }^{2}{\eta }^{2}{\rm{d}}x{\rm{d}}t+\frac{1}{2\varepsilon }\underset{{\vartheta }}{\overset{{\vartheta }+\varepsilon }{\displaystyle \int }}\mathop{\displaystyle \int }\limits_{{B}_{\varrho }}{| {\tau }_{s,h}u| }^{2}{\eta }^{2}{\rm{d}}x{\rm{d}}t.\end{array}By using the estimate 12(ϱ2−r2)≤1(ϱ−r)2\frac{1}{2\left({\varrho }^{2}-{r}^{2})}\le \frac{1}{{\left(\varrho -r)}^{2}}, we obtain (3.5)I+II≔12ε∫ϑϑ+ε∫Bϱ∣τs,hu∣2η2dxdt+∫Qϱ⟨τs,hDf(Du),τs,hDu⟩η2ζεdz≤1(ϱ−r)2∫−ϱ2−r2∫Bϱ∣τs,hu∣2η2dxdt−2∫Qϱ⟨τs,hDf(Du),τs,hu⊗Dη⟩ηζεdz≕III+IV,\begin{array}{rcl}I+II& := & \frac{1}{2\varepsilon }\underset{{\vartheta }}{\overset{{\vartheta }+\varepsilon }{\displaystyle \int }}\mathop{\displaystyle \int }\limits_{{B}_{\varrho }}{| {\tau }_{s,h}u| }^{2}{\eta }^{2}{\rm{d}}x{\rm{d}}t+\mathop{\displaystyle \int }\limits_{{Q}_{\varrho }}\langle {\tau }_{s,h}Df\left(Du),{\tau }_{s,h}Du\rangle {\eta }^{2}{\zeta }_{\varepsilon }{\rm{d}}z\\ & \le & \frac{1}{{\left(\varrho -r)}^{2}}\underset{-{\varrho }^{2}}{\overset{-{r}^{2}}{\displaystyle \int }}\mathop{\displaystyle \int }\limits_{{B}_{\varrho }}{| {\tau }_{s,h}u| }^{2}{\eta }^{2}{\rm{d}}x{\rm{d}}t-2\mathop{\displaystyle \int }\limits_{{Q}_{\varrho }}\langle {\tau }_{s,h}Df\left(Du),{\tau }_{s,h}u\displaystyle \otimes D\eta \rangle \eta {\zeta }_{\varepsilon }{\rm{d}}z\hspace{0.33em}=: \hspace{0.33em}III+IV,\end{array}with the obvious meaning of II–IVIV. Due to the properties of the cutoff function η\eta , we have I≥12ε∫ϑϑ+ε∫Br∣τs,hu∣2dxdtI\ge \frac{1}{2\varepsilon }\underset{{\vartheta }}{\overset{{\vartheta }+\varepsilon }{\int }}\mathop{\int }\limits_{{B}_{r}}{| {\tau }_{s,h}u| }^{2}{\rm{d}}x{\rm{d}}tand ∣III∣≤1(ϱ−r)2∫Qϱ∣τs,hu∣2dz.| III| \le \frac{1}{{\left(\varrho -r)}^{2}}\mathop{\int }\limits_{{Q}_{\varrho }}{| {\tau }_{s,h}u| }^{2}{\rm{d}}z.The second term can be rewritten in the following way: II=∫Qϱ∫01ddα[Df(Du+ατs,hDu)],τs,hDuη2ζεdαdz=∫Qϱ∫01⟨D2f(Du+ατs,hDu)τs,hDu,τs,hDu⟩η2ζεdαdz.\begin{array}{rcl}II& =& \mathop{\displaystyle \int }\limits_{{Q}_{\varrho }}\underset{0}{\overset{1}{\displaystyle \int }}\left\langle \frac{{\rm{d}}}{{\rm{d}}\alpha }{[}Df\left(Du+\alpha {\tau }_{s,h}Du)],{\tau }_{s,h}Du\right\rangle {\eta }^{2}{\zeta }_{\varepsilon }{\rm{d}}\alpha {\rm{d}}z\\ & =& \mathop{\displaystyle \int }\limits_{{Q}_{\varrho }}\underset{0}{\overset{1}{\displaystyle \int }}\langle {D}^{2}f\left(Du+\alpha {\tau }_{s,h}Du){\tau }_{s,h}Du,{\tau }_{s,h}Du\rangle {\eta }^{2}{\zeta }_{\varepsilon }{\rm{d}}\alpha {\rm{d}}z.\end{array}Similarly, the fourth term can be rewritten in the following way: IV=∫Qϱ∫01⟨D2f(Du+ατs,hDu)τs,hDu,τs,hu⊗Dη⟩ηζεdαdz.IV=\mathop{\int }\limits_{{Q}_{\varrho }}\underset{0}{\overset{1}{\int }}\langle {D}^{2}f\left(Du+\alpha {\tau }_{s,h}Du){\tau }_{s,h}Du,{\tau }_{s,h}u\otimes D\eta \rangle \eta {\zeta }_{\varepsilon }{\rm{d}}\alpha {\rm{d}}z.Now we use the Cauchy-Schwarz inequality for the symmetric bilinear form A(σ,σ˜)≔⟨D2f(Du+ατs,hDu)σ,σ˜⟩{\mathcal{A}}\left(\sigma ,\tilde{\sigma }):= \langle {D}^{2}f\left(Du+\alpha {\tau }_{s,h}Du)\sigma ,\tilde{\sigma }\rangle and Young’s inequality to obtain ∣IV∣≤∫Qϱ∫01A(τs,hDu,τs,hDu)A(τs,hu⊗Dη,τs,hu⊗Dη)ηζεdαdz≤12II+12∫Qϱ∫01⟨D2f(Du+ατs,hDu)τs,hu⊗Dη,τs,hu⊗Dη⟩ζεdαdz≤12II+2L(ϱ−r)2∫Qϱ∫011+∑i=1n∣Diu+ατs,hDiu∣pi−2∣τs,hu∣2dαdz,\begin{array}{rcl}| IV| & \le & \mathop{\displaystyle \int }\limits_{{Q}_{\varrho }}\underset{0}{\overset{1}{\displaystyle \int }}\sqrt{{\mathcal{A}}\left({\tau }_{s,h}Du,{\tau }_{s,h}Du)}\sqrt{{\mathcal{A}}\left({\tau }_{s,h}u\displaystyle \otimes D\eta ,{\tau }_{s,h}u\displaystyle \otimes D\eta )}\eta {\zeta }_{\varepsilon }{\rm{d}}\alpha {\rm{d}}z\\ & \le & \frac{1}{2}II+\frac{1}{2}\mathop{\displaystyle \int }\limits_{{Q}_{\varrho }}\underset{0}{\overset{1}{\displaystyle \int }}\langle {D}^{2}f\left(Du+\alpha {\tau }_{s,h}Du){\tau }_{s,h}u\displaystyle \otimes D\eta ,{\tau }_{s,h}u\displaystyle \otimes D\eta \rangle {\zeta }_{\varepsilon }{\rm{d}}\alpha {\rm{d}}z\\ & \le & \frac{1}{2}II+\frac{2L}{{\left(\varrho -r)}^{2}}\mathop{\displaystyle \int }\limits_{{Q}_{\varrho }}\underset{0}{\overset{1}{\displaystyle \int }}\left(1+\mathop{\displaystyle \sum }\limits_{i=1}^{n}{| {D}_{i}u+\alpha {\tau }_{s,h}{D}_{i}u| }^{{p}_{i}-2}\right){| {\tau }_{s,h}u| }^{2}{\rm{d}}\alpha {\rm{d}}z,\end{array}where we used the estimate (3.3) for ∣Dη∣| D\eta | , the growth condition (1.2)2{}_{2}, and the fact that ζε≤1{\zeta }_{\varepsilon }\le 1. By inserting the estimates for the terms II–IVIVinto (3.5) and absorbing the term 12II\frac{1}{2}IIon the left-hand side, we obtain 12ε∫ϑϑ+ε∫Br∣τs,hu∣2dxdt+12∫Qϱ∫01⟨D2f(Du+ατs,hDu)τs,hDu,τs,hDu⟩η2ζεdαdz≤C(L)(ϱ−r)2∫Qϱ∫011+∑i=1n∣Diu+ατs,hDiu∣pi−2∣τs,hu∣2dαdz≤C(L)(ϱ−r)2∑i=1n∫Qϱ∫01(1+∣Diu+ατs,hDiu∣pi−2)dα∣τs,hu∣2dz.\begin{array}{l}\frac{1}{2\varepsilon }\underset{{\vartheta }}{\overset{{\vartheta }+\varepsilon }{\displaystyle \int }}\mathop{\displaystyle \int }\limits_{{B}_{r}}{| {\tau }_{s,h}u| }^{2}{\rm{d}}x{\rm{d}}t+\frac{1}{2}\mathop{\displaystyle \int }\limits_{{Q}_{\varrho }}\underset{0}{\overset{1}{\displaystyle \int }}\langle {D}^{2}f\left(Du+\alpha {\tau }_{s,h}Du){\tau }_{s,h}Du,{\tau }_{s,h}Du\rangle {\eta }^{2}{\zeta }_{\varepsilon }{\rm{d}}\alpha {\rm{d}}z\\ \hspace{1.0em}\le \frac{C\left(L)}{{\left(\varrho -r)}^{2}}\mathop{\displaystyle \int }\limits_{{Q}_{\varrho }}\underset{0}{\overset{1}{\displaystyle \int }}\left(1+\mathop{\displaystyle \sum }\limits_{i=1}^{n}{| {D}_{i}u+\alpha {\tau }_{s,h}{D}_{i}u| }^{{p}_{i}-2}\right){| {\tau }_{s,h}u| }^{2}{\rm{d}}\alpha {\rm{d}}z\\ \hspace{1.0em}\le \frac{C\left(L)}{{\left(\varrho -r)}^{2}}\mathop{\displaystyle \sum }\limits_{i=1}^{n}\mathop{\displaystyle \int }\limits_{{Q}_{\varrho }}\underset{0}{\overset{1}{\displaystyle \int }}(1+{| {D}_{i}u+\alpha {\tau }_{s,h}{D}_{i}u| }^{{p}_{i}-2}){\rm{d}}\alpha {| {\tau }_{s,h}u| }^{2}{\rm{d}}z.\end{array}Now we use the ellipticity condition (1.2)3{}_{3}and apply Lemma 2.2 (with A=Diu(x,t)A={D}_{i}u\left(x,t), B=Diu(x+hes,t)B={D}_{i}u\left(x+h{e}_{s},t), σ=pi−22\sigma =\frac{{p}_{i}-2}{2}, and μ=0\mu =0) to estimate the second term on the left-hand side from below: ∫Qϱ∫01⟨D2f(Du+ατs,hDu)τs,hDu,τs,hDu⟩η2ζεdαdz≥ν∫Qϱ∫01∑i=1n∣Diu+ατs,hDiu∣pi−2∣τs,hDiu∣2η2ζεdαdz≥νC(pi)∑i=1n∫Qϱ(∣Diu(x,t)∣2+∣Diu(x+hes,t)∣2)pi−22∣τs,hDiu∣2η2ζεdz≥νC(pi)∑i=1n∫Qϱ∣τs,hDiu∣piη2ζεdz≥νC(pi)∑i=1n∫Br×(−r2,ϑ)∣τs,hDiu∣pidz.\begin{array}{l}\mathop{\displaystyle \int }\limits_{{Q}_{\varrho }}\underset{0}{\overset{1}{\displaystyle \int }}\langle {D}^{2}f\left(Du+\alpha {\tau }_{s,h}Du){\tau }_{s,h}Du,{\tau }_{s,h}Du\rangle {\eta }^{2}{\zeta }_{\varepsilon }{\rm{d}}\alpha {\rm{d}}z\\ \hspace{1.0em}\ge \nu \mathop{\displaystyle \int }\limits_{{Q}_{\varrho }}\underset{0}{\overset{1}{\displaystyle \int }}\left(\mathop{\displaystyle \sum }\limits_{i=1}^{n}{| {D}_{i}u+\alpha {\tau }_{s,h}{D}_{i}u| }^{{p}_{i}-2}{| {\tau }_{s,h}{D}_{i}u| }^{2}\right){\eta }^{2}{\zeta }_{\varepsilon }{\rm{d}}\alpha {\rm{d}}z\\ \hspace{1.0em}\ge \frac{\nu }{C\left({p}_{i})}\mathop{\displaystyle \sum }\limits_{i=1}^{n}\mathop{\displaystyle \int }\limits_{{Q}_{\varrho }}{({| {D}_{i}u\left(x,t)| }^{2}+{| {D}_{i}u\left(x+h{e}_{s},t)| }^{2})}^{\tfrac{{p}_{i}-2}{2}}{| {\tau }_{s,h}{D}_{i}u| }^{2}{\eta }^{2}{\zeta }_{\varepsilon }{\rm{d}}z\\ \hspace{1.0em}\ge \frac{\nu }{C\left({p}_{i})}\mathop{\displaystyle \sum }\limits_{i=1}^{n}\mathop{\displaystyle \int }\limits_{{Q}_{\varrho }}{| {\tau }_{s,h}{D}_{i}u| }^{{p}_{i}}{\eta }^{2}{\zeta }_{\varepsilon }{\rm{d}}z\\ \hspace{1.0em}\ge \frac{\nu }{C\left({p}_{i})}\mathop{\displaystyle \sum }\limits_{i=1}^{n}\mathop{\displaystyle \int }\limits_{{B}_{r}\times \left(-{r}^{2},{\vartheta })}{| {\tau }_{s,h}{D}_{i}u| }^{{p}_{i}}{\rm{d}}z.\end{array}In the last step, we also used the definition of η\eta and the definition of ζε{\zeta }_{\varepsilon }from (3.4). On the other hand, due to Lemma 2.2 (applied with μ=1\mu =1and A,B,σA,B,\sigma as mentioned earlier) we also have ∫01(1+∣Diu+ατs,hDiu∣pi−2)dα≤2∫01(1+∣Diu+ατs,hDiu∣2)pi−22dα≤C(pi)(1+∣Diu(x,t)∣2+∣Diu(x+hes,t)∣2)pi−22≤C(pi)(1+∣Diu(x,t)∣2+∣Diu(x+hes,t)−Diu(x,t)∣2)pi−22≤C(pi)(1+∣Diu∣+∣τs,hDiu∣)pi−2.\begin{array}{rcl}\underset{0}{\overset{1}{\displaystyle \int }}(1+{| {D}_{i}u+\alpha {\tau }_{s,h}{D}_{i}u| }^{{p}_{i}-2}){\rm{d}}\alpha & \le & 2\underset{0}{\overset{1}{\displaystyle \int }}{(1+{| {D}_{i}u+\alpha {\tau }_{s,h}{D}_{i}u| }^{2})}^{\tfrac{{p}_{i}-2}{2}}{\rm{d}}\alpha \\ & \le & C\left({p}_{i}){(1+{| {D}_{i}u\left(x,t)| }^{2}+{| {D}_{i}u\left(x+h{e}_{s},t)| }^{2})}^{\tfrac{{p}_{i}-2}{2}}\\ & \le & C\left({p}_{i}){(1+{| {D}_{i}u\left(x,t)| }^{2}+{| {D}_{i}u\left(x+h{e}_{s},t)-{D}_{i}u\left(x,t)| }^{2})}^{\tfrac{{p}_{i}-2}{2}}\\ & \le & C\left({p}_{i}){(1+| {D}_{i}u| +| {\tau }_{s,h}{D}_{i}u| )}^{{p}_{i}-2}.\end{array}By putting the previous estimates together, we obtain 1ε∫ϑϑ+ε∫Br∣τs,hu∣2dxdt+∑i=1n∫Br×(−r2,ϑ)∣τs,hDiu∣pidz≤C(L,ν,pi)(ϱ−r)2∑i=1n∫Qϱ(1+∣Diu∣+∣τs,hDiu∣)pi−2∣τs,hu∣2dz.\frac{1}{\varepsilon }\underset{{\vartheta }}{\overset{{\vartheta }+\varepsilon }{\int }}\mathop{\int }\limits_{{B}_{r}}{| {\tau }_{s,h}u| }^{2}{\rm{d}}x{\rm{d}}t+\mathop{\sum }\limits_{i=1}^{n}\mathop{\int }\limits_{{B}_{r}\times \left(-{r}^{2},{\vartheta })}{| {\tau }_{s,h}{D}_{i}u| }^{{p}_{i}}{\rm{d}}z\le \frac{C\left(L,\nu ,{p}_{i})}{{\left(\varrho -r)}^{2}}\mathop{\sum }\limits_{i=1}^{n}\mathop{\int }\limits_{{Q}_{\varrho }}{\left(1+| {D}_{i}u| +| {\tau }_{s,h}{D}_{i}u| )}^{{p}_{i}-2}{| {\tau }_{s,h}u| }^{2}{\rm{d}}z.The first term on the left-hand side converges to ∫Br∣τs,hu(⋅,ϑ)∣2dx{\int }_{{B}_{r}}{| {\tau }_{s,h}u\left(\cdot ,{\vartheta })| }^{2}{\rm{d}}xas ε→0\varepsilon \to 0. Hence, going to the supremum with respect to ϑ{\vartheta }in the first term on the left-hand side, we obtain that supϑ∈(−r2,0)∫Br∣τs,hu(⋅,ϑ)∣2dx≤C(L,ν,pi)(ϱ−r)2∑i=1n∫Qϱ(1+∣Diu∣+∣τs,hDiu∣)pi−2∣τs,hu∣2dz.\mathop{\sup }\limits_{{\vartheta }\in \left(-{r}^{2},0)}\mathop{\int }\limits_{{B}_{r}}{| {\tau }_{s,h}u\left(\cdot ,{\vartheta })| }^{2}{\rm{d}}x\le \frac{C\left(L,\nu ,{p}_{i})}{{\left(\varrho -r)}^{2}}\mathop{\sum }\limits_{i=1}^{n}\mathop{\int }\limits_{{Q}_{\varrho }}{\left(1+| {D}_{i}u| +| {\tau }_{s,h}{D}_{i}u| )}^{{p}_{i}-2}{| {\tau }_{s,h}u| }^{2}{\rm{d}}z.In the second term on the left-hand side, we let ϑ→0{\vartheta }\to 0to obtain ∑i=1n∫Br×(−r2,0)∣τs,hDiu∣pidz≤C(L,ν,pi)(ϱ−r)2∑i=1n∫Qϱ(1+∣Diu∣+∣τs,hDiu∣)pi−2∣τs,hu∣2dz.\mathop{\sum }\limits_{i=1}^{n}\mathop{\int }\limits_{{B}_{r}\times \left(-{r}^{2},0)}{| {\tau }_{s,h}{D}_{i}u| }^{{p}_{i}}{\rm{d}}z\le \frac{C\left(L,\nu ,{p}_{i})}{{\left(\varrho -r)}^{2}}\mathop{\sum }\limits_{i=1}^{n}\mathop{\int }\limits_{{Q}_{\varrho }}{\left(1+| {D}_{i}u| +| {\tau }_{s,h}{D}_{i}u| )}^{{p}_{i}-2}{| {\tau }_{s,h}u| }^{2}{\rm{d}}z.By adding the two previous inequalities, we obtain the desired inequality □supϑ∈(−r2,0)∫Br∣τs,hu(⋅,ϑ)∣2dx+∑i=1n∫Qr∣τs,hDiu∣pidz≤C(L,ν,pi)(ϱ−r)2∑i=1n∫Qϱ(1+∣Diu∣+∣τs,hDiu∣)pi−2∣τs,hu∣2dz.\mathop{\sup }\limits_{{\vartheta }\in \left(-{r}^{2},0)}\mathop{\int }\limits_{{B}_{r}}{| {\tau }_{s,h}u\left(\cdot ,{\vartheta })| }^{2}{\rm{d}}x+\mathop{\sum }\limits_{i=1}^{n}\mathop{\int }\limits_{{Q}_{r}}{| {\tau }_{s,h}{D}_{i}u| }^{{p}_{i}}{\rm{d}}z\le \frac{C\left(L,\nu ,{p}_{i})}{{\left(\varrho -r)}^{2}}\mathop{\sum }\limits_{i=1}^{n}\mathop{\int }\limits_{{Q}_{\varrho }}{\left(1+| {D}_{i}u| +| {\tau }_{s,h}{D}_{i}u| )}^{{p}_{i}-2}{| {\tau }_{s,h}u| }^{2}{\rm{d}}z.4Proof of the main theoremWe have now acquired all the necessary tools to prove Theorem 1.4. The main idea is to derive a uniform bound (with respect to hh) for the right-hand side of the Caccioppoli inequality (3.1). Lemma 2.12 then asserts that uubelongs to certain fractional parabolic Sobolev spaces. Subsequently, we can use Lemma 2.11 to improve the integrability, of a fixed partial derivative Diu{D}_{i}u. Finally, we need to perform an iteration procedure to obtain the full higher integrability, i.e., Du∈Llocpn+εDu\in {L}_{{\rm{loc}}}^{{p}_{n}+\varepsilon }.Proof of Theorem 1.4Let z0∈ΩT{z}_{0}\in {\Omega }_{T}, let ϱ∈(0,1)\varrho \in \left(0,1)be some radius such that Q2ϱ(z0)⊂ΩT{Q}_{2\varrho }\left({z}_{0})\subset {\Omega }_{T}, and let ϱ2≤ϱ1<ϱ2≤ϱ\frac{\varrho }{2}\le {\varrho }_{1}\lt {\varrho }_{2}\le \varrho . Furthermore, let h∈(−ϱ,ϱ)h\in \left(-\varrho ,\varrho )and let s∈{1,…,n}s\in \{1,\ldots ,n\}be arbitrary but fixed. In the following, we suppress the center of the cylinder in our notation by writing, e.g., Qϱ{Q}_{\varrho }instead of Qϱ(z0){Q}_{\varrho }\left({z}_{0}). Since Q2ϱ⊂ΩT{Q}_{2\varrho }\subset {\Omega }_{T}and ∣h∣<ϱ| h| \lt \varrho , we can apply Lemma 3.1 with ϱ2+ϱ12,ϱ1\left(\frac{{\varrho }_{2}+{\varrho }_{1}}{2},{\varrho }_{1}\right)instead of (ϱ,r)\left(\varrho ,r), which yields the following estimate: (4.1)supt∈(−ϱ12,0)∫Bϱ1∣τs,hu(⋅,t)∣2dx+∑i=1n∫Qϱ1∣τs,hDiu∣pidz≤C(L,ν,pi)(ϱ2−ϱ1)2∑i=1n∫Qϱ2+ϱ12(1+∣Diu∣+∣τs,hDiu∣)pi−2∣τs,hu∣2dz.\mathop{\sup }\limits_{t\in \left(-{\varrho }_{1}^{2},0)}\mathop{\int }\limits_{{B}_{{\varrho }_{1}}}{| {\tau }_{s,h}u\left(\cdot ,t)| }^{2}{\rm{d}}x+\mathop{\sum }\limits_{i=1}^{n}\mathop{\int }\limits_{{Q}_{{\varrho }_{1}}}{| {\tau }_{s,h}{D}_{i}u| }^{{p}_{i}}{\rm{d}}z\le \frac{C\left(L,\nu ,{p}_{i})}{{\left({\varrho }_{2}-{\varrho }_{1})}^{2}}\mathop{\sum }\limits_{i=1}^{n}\mathop{\int }\limits_{{Q}_{\tfrac{{\varrho }_{2}+{\varrho }_{1}}{2}}}{\left(1+| {D}_{i}u| +| {\tau }_{s,h}{D}_{i}u| )}^{{p}_{i}-2}{| {\tau }_{s,h}u| }^{2}{\rm{d}}z.Step 1: Let us assume that Du∈Llocσ(ΩT,RN×n)for someσ∈[p1,pn),Du\in {L}_{\hspace{0.1em}\text{loc}\hspace{0.1em}}^{\sigma }({\Omega }_{T},{{\mathbb{R}}}^{N\times n})\hspace{1.0em}\hspace{0.1em}\text{for some}\hspace{0.1em}\hspace{0.33em}\sigma \in \left[{p}_{1},{p}_{n}),and note that this assumption is satisfied for the choice σ=p1\sigma ={p}_{1}. Now we want to show that the right-hand side of (4.1) can be bounded in terms of some power of ∣h∣| h| , i.e., 1(ϱ2−ϱ1)2∑i=1n∫Qϱ2+ϱ12(1+∣Diu∣+∣τs,hDiu∣)pi−2∣τs,hu∣2dz︸≕RHSi≤C∣h∣2γ\frac{1}{{\left({\varrho }_{2}-{\varrho }_{1})}^{2}}\mathop{\sum }\limits_{i=1}^{n}\mathop{\underbrace{\mathop{\int }\limits_{{Q}_{\tfrac{{\varrho }_{2}+{\varrho }_{1}}{2}}}{\left(1+| {D}_{i}u| +| {\tau }_{s,h}{D}_{i}u| )}^{{p}_{i}-2}{| {\tau }_{s,h}u| }^{2}{\rm{d}}z}}\limits_{=: {\text{RHS}}_{i}}\le C{| h| }^{2\gamma }for some γ∈(0,1)\gamma \in \left(0,1), with a constant CCthat does not depend on hh. We distinguish between two cases: In the case pi≤σ{p}_{i}\le \sigma , we can use Hölder’s inequality to estimate (4.2)RHSi≤∫Qϱ2+ϱ12(1+∣Diu∣+∣τs,hDiu∣)pidzpi−2pi∫Qϱ2+ϱ12∣τs,hu∣pidz2pi≤C(pi)∫Q2ϱ(1+∣Diu∣)pidzpi−2pi∫Q2ϱ∣Dsu∣pidz2pi︸<∞∣h∣2≤C∣h∣2,\begin{array}{rcl}{\text{RHS}}_{i}& \le & {\left(\mathop{\displaystyle \int }\limits_{{Q}_{\tfrac{{\varrho }_{2}+{\varrho }_{1}}{2}}}{\left(1+| {D}_{i}u| +| {\tau }_{s,h}{D}_{i}u| )}^{{p}_{i}}{\rm{d}}z\right)}^{\tfrac{{p}_{i}-2}{{p}_{i}}}{\left(\mathop{\displaystyle \int }\limits_{{Q}_{\tfrac{{\varrho }_{2}+{\varrho }_{1}}{2}}}{| {\tau }_{s,h}u| }^{{p}_{i}}{\rm{d}}z\right)}^{\tfrac{2}{{p}_{i}}}\\ & \le & C\left({p}_{i})\mathop{\underbrace{{\left(\mathop{\displaystyle \int }\limits_{{Q}_{2\varrho }}{\left(1+| {D}_{i}u| )}^{{p}_{i}}{\rm{d}}z\right)}^{\tfrac{{p}_{i}-2}{{p}_{i}}}{\left(\mathop{\displaystyle \int }\limits_{{Q}_{2\varrho }}{| {D}_{s}u| }^{{p}_{i}}{\rm{d}}z\right)}^{\tfrac{2}{{p}_{i}}}}}\limits_{\lt \infty }{| h| }^{2}\\ & \le & C{| h| }^{2},\end{array}where we used (2.1) and (2.2) in the penultimate step. In the last step, we used the fact that Dsu∈Lpi(Q2ϱ){D}_{s}u\in {L}^{{p}_{i}}\left({Q}_{2\varrho })since pi≤σ{p}_{i}\le \sigma and Du∈Llocσ(ΩT)Du\in {L}_{{\rm{loc}}}^{\sigma }\left({\Omega }_{T}). In these calculations, the constant CCdepends on pi{p}_{i}, ‖Diu‖Lpi(ΩT){\Vert {D}_{i}u\Vert }_{{L}^{{p}_{i}}\left({\Omega }_{T})}and ‖Dsu‖Lpi(Q2ϱ){\Vert {D}_{s}u\Vert }_{{L}^{{p}_{i}}\left({Q}_{2\varrho })}, but not on hh. Note that the estimate above also holds in the limit case pi=2{p}_{i}=2.Let us now consider the case pi>σ{p}_{i}\gt \sigma . This case is more difficult to deal with, since we do not know a priori that Dsu∈Lpi(Q2ϱ){D}_{s}u\in {L}^{{p}_{i}}\left({Q}_{2\varrho }). Let (4.3)ai∈0,σp1pi{a}_{i}\in \left(0,\frac{\sigma {p}_{1}}{{p}_{i}}\right)be a parameter, that will be fixed later. We define the exponents q1=pipi−2,q2=σp12ai,q3=σp1pi2(σp1−aipi).{q}_{1}=\frac{{p}_{i}}{{p}_{i}-2},\hspace{1.0em}{q}_{2}=\frac{\sigma {p}_{1}}{2{a}_{i}},\hspace{1.0em}{q}_{3}=\frac{\sigma {p}_{1}{p}_{i}}{2\left(\sigma {p}_{1}-{a}_{i}{p}_{i})}.Note that q1,q2,q3>1{q}_{1},{q}_{2},{q}_{3}\gt 1and 1q1+1q2+1q3=1\frac{1}{{q}_{1}}+\frac{1}{{q}_{2}}+\frac{1}{{q}_{3}}=1. We apply Hölder’s inequality with these exponents and use (2.1) and (2.2) to obtain RHSi=∫Qϱ2+ϱ12(1+∣Diu∣+∣τs,hDiu∣)pi−2∣τs,hu∣2aip1∣τs,hu∣2(p1−ai)p1dz≤∫Qϱ2+ϱ12(1+∣Diu∣+∣τs,hDiu∣)pidzpi−2pi∫Qϱ2+ϱ12∣τs,hu∣σdz2aiσp1∫Qϱ2+ϱ12∣τs,hu∣2q3(p1−ai)p1dz1q3≤C(pi)∫Q2ϱ(1+∣Diu∣)pidzpi−2pi∫Q2ϱ∣Dsu∣σdz2aiσp1∣h∣2aip1∫Qϱ2+ϱ12∣τs,hu∣2q3(p1−ai)p1dz1q3≤C∣h∣2aip1∫Qϱ2+ϱ12∣τs,hu∣2q3(p1−ai)p1dz1q3,\begin{array}{rcl}{\text{RHS}}_{i}& =& \mathop{\displaystyle \int }\limits_{{Q}_{\tfrac{{\varrho }_{2}+{\varrho }_{1}}{2}}}{\left(1+| {D}_{i}u| +| {\tau }_{s,h}{D}_{i}u| )}^{{p}_{i}-2}{| {\tau }_{s,h}u| }^{\tfrac{2{a}_{i}}{{p}_{1}}}{| {\tau }_{s,h}u| }^{\tfrac{2\left({p}_{1}-{a}_{i})}{{p}_{1}}}{\rm{d}}z\\ & \le & {\left(\mathop{\displaystyle \int }\limits_{{Q}_{\tfrac{{\varrho }_{2}+{\varrho }_{1}}{2}}}{\left(1+| {D}_{i}u| +| {\tau }_{s,h}{D}_{i}u| )}^{{p}_{i}}{\rm{d}}z\right)}^{\tfrac{{p}_{i}-2}{{p}_{i}}}{\left(\mathop{\displaystyle \int }\limits_{{Q}_{\tfrac{{\varrho }_{2}+{\varrho }_{1}}{2}}}{| {\tau }_{s,h}u| }^{\sigma }{\rm{d}}z\right)}^{\tfrac{2{a}_{i}}{\sigma {p}_{1}}}{\left(\mathop{\displaystyle \int }\limits_{{Q}_{\tfrac{{\varrho }_{2}+{\varrho }_{1}}{2}}}{| {\tau }_{s,h}u| }^{\frac{2{q}_{3}\left({p}_{1}-{a}_{i})}{{p}_{1}}}{\rm{d}}z\right)}^{\tfrac{1}{{q}_{3}}}\\ & \le & C\left({p}_{i}){\left(\mathop{\displaystyle \int }\limits_{{Q}_{2\varrho }}{\left(1+| {D}_{i}u| )}^{{p}_{i}}{\rm{d}}z\right)}^{\tfrac{{p}_{i}-2}{{p}_{i}}}{\left(\mathop{\displaystyle \int }\limits_{{Q}_{2\varrho }}{| {D}_{s}u| }^{\sigma }{\rm{d}}z\right)}^{\tfrac{2{a}_{i}}{\sigma {p}_{1}}}{| h| }^{\tfrac{2{a}_{i}}{{p}_{1}}}{\left(\mathop{\displaystyle \int }\limits_{{Q}_{\tfrac{{\varrho }_{2}+{\varrho }_{1}}{2}}}{| {\tau }_{s,h}u| }^{\frac{2{q}_{3}\left({p}_{1}-{a}_{i})}{{p}_{1}}}{\rm{d}}z\right)}^{\tfrac{1}{{q}_{3}}}\\ & \le & C{| h| }^{\tfrac{2{a}_{i}}{{p}_{1}}}{\left(\mathop{\displaystyle \int }\limits_{{Q}_{\tfrac{{\varrho }_{2}+{\varrho }_{1}}{2}}}{| {\tau }_{s,h}u| }^{\frac{2{q}_{3}\left({p}_{1}-{a}_{i})}{{p}_{1}}}{\rm{d}}z\right)}^{\tfrac{1}{{q}_{3}}},\end{array}where CCdepends on p1,pi,ai,‖Diu‖Lpi(ΩT){p}_{1},{p}_{i},{a}_{i},{\Vert {D}_{i}u\Vert }_{{L}^{{p}_{i}}\left({\Omega }_{T})}, and ‖Dsu‖Lσ(Q2ϱ){\Vert {D}_{s}u\Vert }_{{L}^{\sigma }\left({Q}_{2\varrho })}, but not on hh. To estimate the remaining integral, we want to apply the parabolic Sobolev embedding from Lemma 2.3. For this, we need to choose ai{a}_{i}in such a way that (4.4)2q3(p1−ai)p1=σ(n+2)n⇔ai=(n+2)σp1−np1pi2pi.\frac{2{q}_{3}\left({p}_{1}-{a}_{i})}{{p}_{1}}=\frac{\sigma \left(n+2)}{n}\hspace{0.33em}\iff \hspace{0.33em}{a}_{i}=\frac{\left(n+2)\sigma {p}_{1}-n{p}_{1}{p}_{i}}{2{p}_{i}}.We have to check that this choice of ai{a}_{i}satisfies the condition (4.3). The upper bound ai<σp1pi{a}_{i}\lt \frac{\sigma {p}_{1}}{{p}_{i}}is satisfied due to pi>σ{p}_{i}\gt \sigma . To verify that the lower bound from (4.3) is satisfied, we use (1.5) to obtain: (n+2)σp1−np1pi≥p1((n+2)p1−npn)>p1(n+2)p1−n(n+2)p1n=0,\left(n+2)\sigma {p}_{1}-n{p}_{1}{p}_{i}\ge {p}_{1}\left(\left(n+2){p}_{1}-n{p}_{n})\gt {p}_{1}\left(\left(n+2){p}_{1}-n\frac{\left(n+2){p}_{1}}{n}\right)=0,and hence, ai>0{a}_{i}\gt 0. Thus, we have proved that (4.4) is an admissible choice for the parameter ai{a}_{i}. We can now apply Lemma 2.3 with (ϱ2,ϱ2+ϱ12)\left({\varrho }_{2},\frac{{\varrho }_{2}+{\varrho }_{1}}{2})instead of (ϱ,r)\left(\varrho ,r)to obtain ∫Qϱ2+ϱ12∣τs,hu∣2q3(p1−ai)p1dz1q3=∫Qϱ2+ϱ12∣τs,hu∣σ(n+2)ndznσ−npi≤C∫Qϱ2∣τs,hDu∣σ+τs,huϱ2−ϱ1σdznσ−npisupt∈(−ϱ22,0)∫Bϱ2∣τs,hu(⋅,t)∣2dx1−σpi≤C(N,n,σ,pi)(ϱ2−ϱ1)n(1−σpi)∫Q2ϱ∣Du∣σdznσ−npisupt∈(−ϱ22,0)∫Bϱ2∣τs,hu(⋅,t)∣2dx1−σpi≤C(N,n,σ,pi,‖Du‖Lσ(Q2ϱ))(ϱ2−ϱ1)n(1−σpi)supt∈(−ϱ22,0)∫Bϱ2∣τs,hu(⋅,t)∣2dx1−σpi.\begin{array}{rcl}{\left(\mathop{\displaystyle \int }\limits_{{Q}_{\tfrac{{\varrho }_{2}+{\varrho }_{1}}{2}}}{| {\tau }_{s,h}u| }^{\frac{2{q}_{3}\left({p}_{1}-{a}_{i})}{{p}_{1}}}{\rm{d}}z\right)}^{\tfrac{1}{{q}_{3}}}& =& {\left(\mathop{\displaystyle \int }\limits_{{Q}_{\tfrac{{\varrho }_{2}+{\varrho }_{1}}{2}}}{| {\tau }_{s,h}u| }^{\frac{\sigma \left(n+2)}{n}}{\rm{d}}z\right)}^{\tfrac{n}{\sigma }-\tfrac{n}{{p}_{i}}}\\ & \le & C{\left(\mathop{\displaystyle \int }\limits_{{Q}_{{\varrho }_{2}}}{| {\tau }_{s,h}Du| }^{\sigma }+{\left|\frac{{\tau }_{s,h}u}{{\varrho }_{2}-{\varrho }_{1}}\right|}^{\sigma }{\rm{d}}z\right)}^{\tfrac{n}{\sigma }-\tfrac{n}{{p}_{i}}}{\left(\mathop{\sup }\limits_{t\in \left(-{\varrho }_{2}^{2},0)}\mathop{\displaystyle \int }\limits_{{B}_{{\varrho }_{2}}}{| {\tau }_{s,h}u\left(\cdot ,t)| }^{2}{\rm{d}}x\right)}^{1-\tfrac{\sigma }{{p}_{i}}}\\ & \le & \frac{C\left(N,n,\sigma ,{p}_{i})}{{\left({\varrho }_{2}-{\varrho }_{1})}^{n\left(1-\tfrac{\sigma }{{p}_{i}})}}{\left(\mathop{\displaystyle \int }\limits_{{Q}_{2\varrho }}{| Du| }^{\sigma }{\rm{d}}z\right)}^{\tfrac{n}{\sigma }-\tfrac{n}{{p}_{i}}}{\left(\mathop{\sup }\limits_{t\in \left(-{\varrho }_{2}^{2},0)}\mathop{\displaystyle \int }\limits_{{B}_{{\varrho }_{2}}}{| {\tau }_{s,h}u\left(\cdot ,t)| }^{2}{\rm{d}}x\right)}^{1-\tfrac{\sigma }{{p}_{i}}}\\ & \le & \frac{C(N,n,\sigma ,{p}_{i},{\Vert Du\Vert }_{{L}^{\sigma }\left({Q}_{2\varrho })})}{{\left({\varrho }_{2}-{\varrho }_{1})}^{n\left(1-\tfrac{\sigma }{{p}_{i}})}}{\left(\mathop{\sup }\limits_{t\in \left(-{\varrho }_{2}^{2},0)}\mathop{\displaystyle \int }\limits_{{B}_{{\varrho }_{2}}}{| {\tau }_{s,h}u\left(\cdot ,t)| }^{2}{\rm{d}}x\right)}^{1-\tfrac{\sigma }{{p}_{i}}}.\end{array}In these calculations, we used the fact that ϱ2−ϱ1≤ϱ2<1{\varrho }_{2}-{\varrho }_{1}\le \frac{\varrho }{2}\lt 1and ∫Qϱ2∣τs,hDu∣σ+∣τs,hu∣σdz≤(2σ+∣h∣σ)∫Q2ϱ∣Du∣σdz≤(2σ+1)∫Q2ϱ∣Du∣σdz.\mathop{\int }\limits_{{Q}_{{\varrho }_{2}}}{| {\tau }_{s,h}Du| }^{\sigma }+{| {\tau }_{s,h}u| }^{\sigma }{\rm{d}}z\le \left({2}^{\sigma }+{| h| }^{\sigma })\mathop{\int }\limits_{{Q}_{2\varrho }}{| Du| }^{\sigma }{\rm{d}}z\le \left({2}^{\sigma }+1)\mathop{\int }\limits_{{Q}_{2\varrho }}{| Du| }^{\sigma }{\rm{d}}z.We combine the previous estimates to obtain RHSi≤C(ϱ2−ϱ1)n(1−σpi)∣h∣2aip1supt∈(−ϱ22,0)∫Bϱ2∣τs,hu(⋅,t)∣2dx1−σpi=C(ϱ2−ϱ1)n(1−σpi)∣h∣(n+2)σ−npipisupt∈(−ϱ22,0)∫Bϱ2∣τs,hu(⋅,t)∣2dx1−σpi,\begin{array}{rcl}{\text{RHS}}_{i}& \le & \frac{C}{{\left({\varrho }_{2}-{\varrho }_{1})}^{n\left(1-\tfrac{\sigma }{{p}_{i}})}}{| h| }^{\tfrac{2{a}_{i}}{{p}_{1}}}{\left(\mathop{\sup }\limits_{t\in \left(-{\varrho }_{2}^{2},0)}\mathop{\displaystyle \int }\limits_{{B}_{{\varrho }_{2}}}{| {\tau }_{s,h}u\left(\cdot ,t)| }^{2}{\rm{d}}x\right)}^{1-\tfrac{\sigma }{{p}_{i}}}\\ & =& \frac{C}{{\left({\varrho }_{2}-{\varrho }_{1})}^{n\left(1-\tfrac{\sigma }{{p}_{i}})}}{| h| }^{\tfrac{\left(n+2)\sigma -n{p}_{i}}{{p}_{i}}}{\left(\mathop{\sup }\limits_{t\in \left(-{\varrho }_{2}^{2},0)}\mathop{\displaystyle \int }\limits_{{B}_{{\varrho }_{2}}}{| {\tau }_{s,h}u\left(\cdot ,t)| }^{2}{\rm{d}}x\right)}^{1-\tfrac{\sigma }{{p}_{i}}},\end{array}where the constant CCdepends on N,n,σ,p1,pi,‖Diu‖Lpi(ΩT)N,n,\sigma ,{p}_{1},{p}_{i},{\Vert {D}_{i}u\Vert }_{{L}^{{p}_{i}}\left({\Omega }_{T})}, and ‖Du‖Lσ(Q2ϱ){\Vert Du\Vert }_{{L}^{\sigma }\left({Q}_{2\varrho })}. We multiply this inequality with 1(ϱ2−ϱ1)2\frac{1}{{\left({\varrho }_{2}-{\varrho }_{1})}^{2}}and apply Young’s inequality with exponents pipi−σ,piσ\frac{{p}_{i}}{{p}_{i}-\sigma },\frac{{p}_{i}}{\sigma }to obtain RHSi(ϱ2−ϱ1)2≤C(ϱ2−ϱ1)n+2−nσpi∣h∣(n+2)σ−npipisupt∈(−ϱ22,0)∫Bϱ2∣τs,hu(⋅,t)∣2dx1−σpi≤12nsupt∈(−ϱ22,0)∫Bϱ2∣τs,hu(⋅,t)∣2dx+C(ϱ2−ϱ1)(n+2)piσ−n∣h∣(n+2)σ−npiσ≤12nsupt∈(−ϱ22,0)∫Bϱ2∣τs,hu(⋅,t)∣2dx+C(ϱ2−ϱ1)4+4n∣h∣(n+2)σ−npiσ,\begin{array}{rcl}\frac{{\text{RHS}}_{i}}{{\left({\varrho }_{2}-{\varrho }_{1})}^{2}}& \le & \frac{C}{{\left({\varrho }_{2}-{\varrho }_{1})}^{n+2-\tfrac{n\sigma }{{p}_{i}}}}{| h| }^{\tfrac{\left(n+2)\sigma -n{p}_{i}}{{p}_{i}}}{\left(\mathop{\sup }\limits_{t\in \left(-{\varrho }_{2}^{2},0)}\mathop{\displaystyle \int }\limits_{{B}_{{\varrho }_{2}}}{| {\tau }_{s,h}u\left(\cdot ,t)| }^{2}{\rm{d}}x\right)}^{1-\tfrac{\sigma }{{p}_{i}}}\\ & \le & \frac{1}{2n}\mathop{\sup }\limits_{t\in \left(-{\varrho }_{2}^{2},0)}\mathop{\displaystyle \int }\limits_{{B}_{{\varrho }_{2}}}{| {\tau }_{s,h}u\left(\cdot ,t)| }^{2}{\rm{d}}x+\frac{C}{{\left({\varrho }_{2}-{\varrho }_{1})}^{\left(n+2)\tfrac{{p}_{i}}{\sigma }-n}}{| h| }^{\tfrac{\left(n+2)\sigma -n{p}_{i}}{\sigma }}\\ & \le & \frac{1}{2n}\mathop{\sup }\limits_{t\in \left(-{\varrho }_{2}^{2},0)}\mathop{\displaystyle \int }\limits_{{B}_{{\varrho }_{2}}}{| {\tau }_{s,h}u\left(\cdot ,t)| }^{2}{\rm{d}}x+\frac{C}{{\left({\varrho }_{2}-{\varrho }_{1})}^{4+\tfrac{4}{n}}}{| h| }^{\tfrac{\left(n+2)\sigma -n{p}_{i}}{\sigma }},\end{array}where CCadmits the same dependencies as specified earlier. In the last step, we used that ϱ2−ϱ1≤ϱ2<1{\varrho }_{2}-{\varrho }_{1}\le \frac{\varrho }{2}\lt 1and piσ≤pnp1<n+2n\frac{{p}_{i}}{\sigma }\le \frac{{p}_{n}}{{p}_{1}}\lt \frac{n+2}{n}due to (1.5). By inserting (4.2) and the last inequality into (4.1) and summing over i=1,…,ni=1,\ldots ,n, we obtain supt∈(−ϱ12,0)∫Bϱ1∣τs,hu(⋅,t)∣2dx+∑i=1n∫Qϱ1∣τs,hDiu∣pidz≤12supt∈(−ϱ22,0)∫Bϱ2∣τs,hu(⋅,t)∣2dx+C(ϱ2−ϱ1)4+4n∑i=1n∣h∣min2,(n+2)σ−npiσ≤12supt∈(−ϱ22,0)∫Bϱ2∣τs,hu(⋅,t)∣2dx+C(ϱ2−ϱ1)4+4n∣h∣(n+2)σ−npnσ.\begin{array}{l}\mathop{\sup }\limits_{t\in \left(-{\varrho }_{1}^{2},0)}\mathop{\displaystyle \int }\limits_{{B}_{{\varrho }_{1}}}{| {\tau }_{s,h}u\left(\cdot ,t)| }^{2}{\rm{d}}x+\mathop{\displaystyle \sum }\limits_{i=1}^{n}\mathop{\displaystyle \int }\limits_{{Q}_{{\varrho }_{1}}}{| {\tau }_{s,h}{D}_{i}u| }^{{p}_{i}}{\rm{d}}z\\ \hspace{1.0em}\le \frac{1}{2}\mathop{\sup }\limits_{t\in \left(-{\varrho }_{2}^{2},0)}\mathop{\displaystyle \int }\limits_{{B}_{{\varrho }_{2}}}{| {\tau }_{s,h}u\left(\cdot ,t)| }^{2}{\rm{d}}x+\frac{C}{{\left({\varrho }_{2}-{\varrho }_{1})}^{4+\tfrac{4}{n}}}\mathop{\displaystyle \sum }\limits_{i=1}^{n}{| h| }^{\min \left\{2,\tfrac{\left(n+2)\sigma -n{p}_{i}}{\sigma }\right\}}\\ \hspace{1.0em}\le \frac{1}{2}\mathop{\sup }\limits_{t\in \left(-{\varrho }_{2}^{2},0)}\mathop{\displaystyle \int }\limits_{{B}_{{\varrho }_{2}}}{| {\tau }_{s,h}u\left(\cdot ,t)| }^{2}{\rm{d}}x+\frac{C}{{\left({\varrho }_{2}-{\varrho }_{1})}^{4+\tfrac{4}{n}}}{| h| }^{\tfrac{\left(n+2)\sigma -n{p}_{n}}{\sigma }}.\end{array}At this point, we can apply Lemma 2.1 with the choices Φ(r)=supt∈(−r2,0)∫Br∣τs,hu(⋅,t)∣2dx+∑i=1n∫Qr∣τs,hDiu∣pidz,A=C∣h∣(n+2)σ−npnσ\begin{array}{rcl}\Phi \left(r)& =& \mathop{\sup }\limits_{t\in \left(-{r}^{2},0)}\mathop{\displaystyle \int }\limits_{{B}_{r}}{| {\tau }_{s,h}u\left(\cdot ,t)| }^{2}{\rm{d}}x+\mathop{\displaystyle \sum }\limits_{i=1}^{n}\mathop{\displaystyle \int }\limits_{{Q}_{r}}{| {\tau }_{s,h}{D}_{i}u| }^{{p}_{i}}{\rm{d}}z,\\ A& =& C{| h| }^{\tfrac{\left(n+2)\sigma -n{p}_{n}}{\sigma }}\end{array}and α=4+4n\alpha =4+\frac{4}{n}to absorb the sup\sup -term into the left-hand side. Thus, we conclude with supt∈(−(ϱ/2)2,0)∫Bϱ/2∣τs,hu(⋅,t)∣2dx+∑i=1n∫Qϱ/2∣τs,hDiu∣pidz≤Cϱ4+4n∣h∣(n+2)σ−npnσ,\mathop{\sup }\limits_{t\in (-{\left(\varrho \text{/}2)}^{2},0)}\mathop{\int }\limits_{{B}_{\varrho \text{/}2}}{| {\tau }_{s,h}u\left(\cdot ,t)| }^{2}{\rm{d}}x+\mathop{\sum }\limits_{i=1}^{n}\mathop{\int }\limits_{{Q}_{\varrho \text{/}2}}{| {\tau }_{s,h}{D}_{i}u| }^{{p}_{i}}{\rm{d}}z\le \frac{C}{{\varrho }^{4+\tfrac{4}{n}}}{| h| }^{\tfrac{\left(n+2)\sigma -n{p}_{n}}{\sigma }},with a constant CCthat does not depend on hh. We also note that (n+2)σ−npnσ≥(n+2)p1−npnσ>0\frac{\left(n+2)\sigma -n{p}_{n}}{\sigma }\ge \frac{\left(n+2){p}_{1}-n{p}_{n}}{\sigma }\gt 0due to (1.5).Step 2: Let us summarize the results from Step 1. Under the assumption that Q2ϱ⊂ΩT{Q}_{2\varrho }\subset {\Omega }_{T}and Du∈Llocσ(ΩT,RN×n)for someσ∈[p1,pn),Du\in {L}_{\hspace{0.1em}\text{loc}\hspace{0.1em}}^{\sigma }({\Omega }_{T},{{\mathbb{R}}}^{N\times n})\hspace{1.0em}\hspace{0.1em}\text{for some}\hspace{0.1em}\hspace{0.33em}\sigma \in \left[{p}_{1},{p}_{n}),we proved that there exists a constant CC(independent of hh) such that for all ∣h∣<ϱ| h| \lt \varrho , we have supt∈(−(ϱ/2)2,0)∫Bϱ/2∣τs,hu(⋅,t)∣2dx+∑i=1n∫Qϱ/2∣τs,hDiu∣pidz≤Cϱ4+4n∣h∣(n+2)σ−npnσ.\mathop{\sup }\limits_{t\in (-{\left(\varrho \text{/}2)}^{2},0)}\mathop{\int }\limits_{{B}_{\varrho \text{/}2}}{| {\tau }_{s,h}u\left(\cdot ,t)| }^{2}{\rm{d}}x+\mathop{\sum }\limits_{i=1}^{n}\mathop{\int }\limits_{{Q}_{\varrho \text{/}2}}{| {\tau }_{s,h}{D}_{i}u| }^{{p}_{i}}{\rm{d}}z\le \frac{C}{{\varrho }^{4+\tfrac{4}{n}}}{| h| }^{\tfrac{\left(n+2)\sigma -n{p}_{n}}{\sigma }}.We are now in the position to apply Lemma 2.12, which asserts that uulies in the following fractional Sobolev spaces for every O⋐Bϱ/2{\mathcal{O}}\hspace{0.33em}\Subset \hspace{0.33em}{B}_{\varrho \text{/}2}: u∈L∞(−(ϱ/2)2,0;Wμ,2(O,RN))∀μ∈(0,μmax),μmax≔(n+2)σ−npn2σu\in {L}^{\infty }(-{\left(\varrho \text{/}2)}^{2},0;\hspace{0.33em}{W}^{\mu ,2}({\mathcal{O}},{{\mathbb{R}}}^{N}))\hspace{1.0em}\forall \mu \in (0,{\mu }_{\max }),\hspace{0.33em}{\mu }_{\max }:= \frac{\left(n+2)\sigma -n{p}_{n}}{2\sigma }and Diu∈Lpi(−(ϱ/2)2,0;Wλ,pi(O,RN))∀λ∈(0,λmax(i)),λmax(i)≔(n+2)σ−npnσpi.{D}_{i}u\in {L}^{{p}_{i}}(-{\left(\varrho \text{/}2)}^{2},0;\hspace{0.33em}{W}^{\lambda ,{p}_{i}}({\mathcal{O}},{{\mathbb{R}}}^{N}))\hspace{1.0em}\forall \lambda \in (0,{\lambda }_{\max }^{\left(i)}),\hspace{0.33em}{\lambda }_{\max }^{\left(i)}:= \frac{\left(n+2)\sigma -n{p}_{n}}{\sigma {p}_{i}}.Since u∈Lp1(−(ϱ/2)2,0;W1,p1(O,RN))u\in {L}^{{p}_{1}}(-{\left(\varrho \text{/}2)}^{2},0;\hspace{0.33em}{W}^{1,{p}_{1}}({\mathcal{O}},{{\mathbb{R}}}^{N}))by definition, we can apply Lemma 2.11 with i=1i=1to obtain D1u∈Ls(O×(−(ϱ/2)2,0),RN){D}_{1}u\in {L}^{s}({\mathcal{O}}\times \left(-{\left(\varrho \text{/}2)}^{2},0),{{\mathbb{R}}}^{N})for all s>p1s\gt {p}_{1}such that (s−p1)1−μmax+n2<λmax(1)p1⇔(s−p1)1−(n+2)σ−npn2σ+n2<(n+2)σ−npnσ⇔(s−p1)npn2σ<(n+2)σ−npnσ⇔s<p1+2(n+2)σ−2npnnpn.\begin{array}{l}\left(s-{p}_{1})\left(1-{\mu }_{\max }+\frac{n}{2}\right)\lt {\lambda }_{\max }^{\left(1)}{p}_{1}\\ \hspace{1.0em}\iff \left(s-{p}_{1})\left(1-\frac{\left(n+2)\sigma -n{p}_{n}}{2\sigma }+\frac{n}{2}\right)\lt \frac{\left(n+2)\sigma -n{p}_{n}}{\sigma }\\ \hspace{1.0em}\iff \left(s-{p}_{1})\frac{n{p}_{n}}{2\sigma }\lt \frac{\left(n+2)\sigma -n{p}_{n}}{\sigma }\\ \hspace{1.0em}\iff s\lt {p}_{1}+\frac{2\left(n+2)\sigma -2n{p}_{n}}{n{p}_{n}}.\end{array}Since the center of the cylinder Qϱ{Q}_{\varrho }was arbitrary, we thus obtain (4.5)D1u∈Llocs(ΩT,RN)∀s<sˆ1(σ)≔p1+2(n+2)σ−2npnnpn.{D}_{1}u\in {L}_{\hspace{0.1em}\text{loc}\hspace{0.1em}}^{s}({\Omega }_{T},{{\mathbb{R}}}^{N})\hspace{1.0em}\forall s\lt {\hat{s}}_{1}\left(\sigma ):= {p}_{1}+\frac{2\left(n+2)\sigma -2n{p}_{n}}{n{p}_{n}}.Step 3: Let us summarize the previous two steps: From the assumption that Du∈Llocσ(ΩT,RN×n)Du\in {L}_{\hspace{0.1em}\text{loc}\hspace{0.1em}}^{\sigma }({\Omega }_{T},{{\mathbb{R}}}^{N\times n})for some σ∈[p1,pn)\sigma \in \left[{p}_{1},{p}_{n}), we have deduced the improved integrability (4.5). We will now show that (4.6)Sˆ≔infσ∈[p1,pn)(sˆ1(σ)−σ)>0.\hat{S}:= \mathop{\inf }\limits_{\sigma \in \left[{p}_{1},{p}_{n})}({\hat{s}}_{1}\left(\sigma )-\sigma )\gt 0.We have sˆ1(σ)−σ=p1+(2(n+2)−npn)σ−2npnnpn.{\hat{s}}_{1}\left(\sigma )-\sigma ={p}_{1}+\frac{\left(2\left(n+2)-n{p}_{n})\sigma -2n{p}_{n}}{n{p}_{n}}.There are three cases to consider (depending on the sign of 2(n+2)−npn2\left(n+2)-n{p}_{n}).Case 1: If 2(n+2)−npn>0,2\left(n+2)-n{p}_{n}\gt 0,we have infσ∈[p1,pn)(sˆ1(σ)−σ)=sˆ1(p1)−p1=2(n+2)p1−2npnnpn>0,\mathop{\inf }\limits_{\sigma \in \left[{p}_{1},{p}_{n})}({\hat{s}}_{1}\left(\sigma )-\sigma )={\hat{s}}_{1}\left({p}_{1})-{p}_{1}=\frac{2\left(n+2){p}_{1}-2n{p}_{n}}{n{p}_{n}}\gt 0,due to (1.5).Case 2: If 2(n+2)−npn=0⇔pn=2(n+2)n=2+4n,2\left(n+2)-n{p}_{n}=0\hspace{0.33em}\iff \hspace{0.33em}{p}_{n}=\frac{2\left(n+2)}{n}=2+\frac{4}{n},we have p1>2{p}_{1}\gt 2, since otherwise the gap condition (1.4) would not hold. Hence, we have infσ∈[p1,pn)(sˆ1(σ)−σ)=sˆ1(p1)−p1=p1−2>0.\mathop{\inf }\limits_{\sigma \in \left[{p}_{1},{p}_{n})}({\hat{s}}_{1}\left(\sigma )-\sigma )={\hat{s}}_{1}\left({p}_{1})-{p}_{1}={p}_{1}-2\gt 0.Case 3: If 2(n+2)−npn<0,2\left(n+2)-n{p}_{n}\lt 0,we have infσ∈[p1,pn)(sˆ1(σ)−σ)=sˆ1(pn)−pn=p1+4n−pn>0,\mathop{\inf }\limits_{\sigma \in \left[{p}_{1},{p}_{n})}({\hat{s}}_{1}\left(\sigma )-\sigma )={\hat{s}}_{1}\left({p}_{n})-{p}_{n}={p}_{1}+\frac{4}{n}-{p}_{n}\gt 0,due to (1.4). Thus, we can conclude that (4.6) holds in all cases.Step 4: Since the integrability gain from σ\sigma to sˆ1(σ){\hat{s}}_{1}\left(\sigma )is uniform with respect to σ\sigma due to (4.6), we can now perform an iteration procedure to show the desired higher integrability. First, we want to show that D1u∈Llocp2(ΩT,RN){D}_{1}u\in {L}_{{\rm{loc}}}^{{p}_{2}}({\Omega }_{T},{{\mathbb{R}}}^{N}). To achieve this, we consider the following iteration scheme (which is possible since Du∈Llocp1(ΩT,RN×n)Du\in {L}_{{\rm{loc}}}^{{p}_{1}}({\Omega }_{T},{{\mathbb{R}}}^{N\times n})by definition): σ1=p1⇒D1u∈Llocp1+Sˆ⇒Du∈Llocmin{p1+Sˆ,p2}σ2=p1+Sˆ⇒D1u∈Llocp1+2Sˆ⇒Du∈Llocmin{p1+2Sˆ,p2}⋮\begin{array}{rclllll}{\sigma }_{1}& =& {p}_{1}& \Rightarrow & {D}_{1}u\in {L}_{{\rm{loc}}}^{{p}_{1}+\hat{S}}& \Rightarrow & Du\in {L}_{{\rm{loc}}}^{\min \{{p}_{1}+\hat{S},{p}_{2}\}}\\ {\sigma }_{2}& =& {p}_{1}+\hat{S}& \Rightarrow & {D}_{1}u\in {L}_{{\rm{loc}}}^{{p}_{1}+2\hat{S}}& \Rightarrow & Du\in {L}_{{\rm{loc}}}^{\min \{{p}_{1}+2\hat{S},{p}_{2}\}}\\ & \vdots & & & & & \end{array}Clearly, we obtain D1u∈Llocp2(ΩT,RN){D}_{1}u\in {L}_{{\rm{loc}}}^{{p}_{2}}\left({\Omega }_{T},{{\mathbb{R}}}^{N}), and hence, Du∈Llocp2(ΩT,RN×n)Du\in {L}_{{\rm{loc}}}^{{p}_{2}}\left({\Omega }_{T},{{\mathbb{R}}}^{N\times n})after finitely many iterations. The next step in the iteration is to show that D1u,D2u∈Llocp3{D}_{1}u,{D}_{2}u\in {L}_{{\rm{loc}}}^{{p}_{3}}. To achieve this, we apply Lemma (2.11) with p2{p}_{2}instead of ppto obtain that D2u∈Llocs(ΩT,RN){D}_{2}u\in {L}_{{\rm{loc}}}^{s}({\Omega }_{T},{{\mathbb{R}}}^{N})for all s>p2s\gt {p}_{2}such that (s−p2)1−μmax+n2<λmax(2)p2⇔s<p2+2(n+2)σ−2npnnpn≕sˆ2(σ).\left(s-{p}_{2})\left(1-{\mu }_{\max }+\frac{n}{2}\right)\lt {\lambda }_{\max }^{\left(2)}{p}_{2}\iff s\lt {p}_{2}+\frac{2\left(n+2)\sigma -2n{p}_{n}}{n{p}_{n}}\hspace{0.33em}=: \hspace{0.33em}{\hat{s}}_{2}\left(\sigma ).Due to p2≥p1{p}_{2}\ge {p}_{1}, we have sˆ2(σ)−σ≥sˆ1(σ)−σ≥Sˆ{\hat{s}}_{2}\left(\sigma )-\sigma \ge {\hat{s}}_{1}\left(\sigma )-\sigma \ge \hat{S}for all σ∈[p2,pn)\sigma \in \left[{p}_{2},{p}_{n}), which means that we gain at least as much integrability for D2u{D}_{2}uas for D1u{D}_{1}u. We can now use the following iteration scheme to simultaneously improve the integrability for D1u{D}_{1}uand D2u{D}_{2}u: σ1=p2⇒D1u,D2u∈Llocp2+Sˆ⇒Du∈Llocmin{p2+Sˆ,p3}σ2=p2+Sˆ⇒D1u,D2u∈Llocp2+2Sˆ⇒Du∈Llocmin{p2+2Sˆ,p3}⋮\begin{array}{rclllll}{\sigma }_{1}& =& {p}_{2}& \Rightarrow & {D}_{1}u,{D}_{2}u\in {L}_{{\rm{loc}}}^{{p}_{2}+\hat{S}}& \Rightarrow & Du\in {L}_{{\rm{loc}}}^{\min \{{p}_{2}+\hat{S},{p}_{3}\}}\\ {\sigma }_{2}& =& {p}_{2}+\hat{S}& \Rightarrow & {D}_{1}u,{D}_{2}u\in {L}_{{\rm{loc}}}^{{p}_{2}+2\hat{S}}& \Rightarrow & Du\in {L}_{{\rm{loc}}}^{\min \{{p}_{2}+2\hat{S},{p}_{3}\}}\\ & \vdots & & & & & \end{array}After finitely many steps of this iteration, we obtain Du∈Llocp3(ΩT,RN×n)Du\in {L}_{{\rm{loc}}}^{{p}_{3}}({\Omega }_{T},{{\mathbb{R}}}^{N\times n}). In an analogous way, we can improve the integrability of D1u,D2u,D3u{D}_{1}u,{D}_{2}u,{D}_{3}uto obtain Du∈Llocp4(ΩT,RN×n)Du\in {L}_{{\rm{loc}}}^{{p}_{4}}({\Omega }_{T},{{\mathbb{R}}}^{N\times n}). Inductively we obtain Du∈Llocpn(ΩT,RN×n)Du\in {L}_{{\rm{loc}}}^{{p}_{n}}({\Omega }_{T},{{\mathbb{R}}}^{N\times n}). Due to (4.5), the maximum amount of integrability that can be obtained for D1u{D}_{1}u(and hence for DuDu) is limited by limσ→pnsˆ1(σ)=p1+4n.\mathop{\mathrm{lim}}\limits_{\sigma \to {p}_{n}}{\hat{s}}_{1}\left(\sigma )={p}_{1}+\frac{4}{n}.This implies that we obtain Du∈LlocsDu\in {L}_{{\rm{loc}}}^{s}for all s<p1+4ns\lt {p}_{1}+\frac{4}{n}. This proves the assertion of the theorem.□

Journal

Advances in Nonlinear Analysisde Gruyter

Published: Jan 1, 2023

Keywords: finite differences; higher integrability; parabolic systems; Primary: 35K40; Secondary: 35B65

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