XII Symposium of Probability and Stochastic ProcessesA Criterion for Blow Up in Finite Time of a System of 1-Dimensional Reaction-Diffusion Equations

XII Symposium of Probability and Stochastic Processes: A Criterion for Blow Up in Finite Time of... [We give a criterion for blow up in finite time of the system of semilinear partial differential equations ∂ui(t,x)∂t=12∂2uit,x∂x2+φi′xφix∂uit,x∂x+uj1+βit,x\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\frac {\partial u_{i}(t,x)}{\partial t}=\frac {1}{2}\frac {\partial ^{2}u_{i}\left (t,x\right )}{\partial x^{2}}+\frac {\varphi ^{\prime }_{i}\left (x\right )} {\varphi _{i}\left (x\right )}\frac {\partial u_{i}\left (t,x\right )}{\partial x}+u_{j}^{1+\beta _{i}}\left (t,x\right )$$ \end{document}, t > 0, x∈ℝ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$x\in \mathbb {R}$$ \end{document}, with initial values of the form ui0,x=hix∕φix\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$u_{i}\left (0,x\right )={h_{i}\left (x\right )}/{\varphi _{i}\left (x\right )}$$ \end{document}, where 0<φi∈L2ℝ,dx∩C2ℝ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$0<\varphi _{i}\in L^{2}\left (\mathbb {R},\mathrm {d} x\right )\cap C^{2}\left (\mathbb {R}\right )$$ \end{document}, 0≤hi∈L2ℝ,dx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$0\leq h_{i}\in L^{2}\left (\mathbb {R},\mathrm {d} x\right )$$ \end{document}, βi > 0 and i = 1, 2, j = 3 − i. Moreover, we find an upper bound T∗ for the blowup time of such system which depends both on the initial values f1, f2, and the measures μi(dx)=φi2(x)dx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mu _i(\mathrm {d} x)=\varphi _i^2(x)\,\mathrm {d} x$$ \end{document}, i = 1, 2.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

XII Symposium of Probability and Stochastic ProcessesA Criterion for Blow Up in Finite Time of a System of 1-Dimensional Reaction-Diffusion Equations

Part of the Progress in Probability Book Series (volume 73)
Editors: Hernández-Hernández, Daniel; Pardo, Juan Carlos; Rivero, Victor
10 pages

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References (13)

Publisher
Springer International Publishing
© Springer International Publishing AG, part of Springer Nature 2018
ISBN
978-3-319-77642-2
Pages
207 –217
DOI
10.1007/978-3-319-77643-9_7
Publisher site
See Chapter on Publisher Site

Abstract

[We give a criterion for blow up in finite time of the system of semilinear partial differential equations ∂ui(t,x)∂t=12∂2uit,x∂x2+φi′xφix∂uit,x∂x+uj1+βit,x\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\frac {\partial u_{i}(t,x)}{\partial t}=\frac {1}{2}\frac {\partial ^{2}u_{i}\left (t,x\right )}{\partial x^{2}}+\frac {\varphi ^{\prime }_{i}\left (x\right )} {\varphi _{i}\left (x\right )}\frac {\partial u_{i}\left (t,x\right )}{\partial x}+u_{j}^{1+\beta _{i}}\left (t,x\right )$$ \end{document}, t > 0, x∈ℝ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$x\in \mathbb {R}$$ \end{document}, with initial values of the form ui0,x=hix∕φix\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$u_{i}\left (0,x\right )={h_{i}\left (x\right )}/{\varphi _{i}\left (x\right )}$$ \end{document}, where 0<φi∈L2ℝ,dx∩C2ℝ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$0<\varphi _{i}\in L^{2}\left (\mathbb {R},\mathrm {d} x\right )\cap C^{2}\left (\mathbb {R}\right )$$ \end{document}, 0≤hi∈L2ℝ,dx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$0\leq h_{i}\in L^{2}\left (\mathbb {R},\mathrm {d} x\right )$$ \end{document}, βi > 0 and i = 1, 2, j = 3 − i. Moreover, we find an upper bound T∗ for the blowup time of such system which depends both on the initial values f1, f2, and the measures μi(dx)=φi2(x)dx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mu _i(\mathrm {d} x)=\varphi _i^2(x)\,\mathrm {d} x$$ \end{document}, i = 1, 2.]

Published: Jun 27, 2018

Keywords: Semilinear system of PDEs; Local mild solution; Finite time blow up; Primary 60H30; 35K57; 35B35; 60J57