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This article explores a quasilinear wave equation of p-Laplacian type: utt-Δpu-Δut=|u|r-1u\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} u_{tt}-\Delta _{p}u-\Delta u_t=|u|^{r-1}u \end{aligned}$$\end{document}in a bounded domain Ω⊂R3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Omega \subset \mathbb {R}^{3}$$\end{document} and subject to Dirichlet boundary conditions. The operator Δp\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Delta _{p}$$\end{document} denotes the classical p-Laplacian with 2<p<3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$2<p<3$$\end{document} in this paper. We obtain an energy estimate for the solutions by applying a new method. Moreover, a blow-up result is proved for solutions with arbitrarily positive initial energy. An estimate of the lifespan of the solutions is showed as well. These results give some answers to the unsolved problems in Pei et al. (J Math Phys 56:081503, 2015). Besides, we also provide some numerical examples to illustrate our results.
Analysis and Mathematical Physics – Springer Journals
Published: Jun 1, 2023
Keywords: p-Laplacian; Arbitrarily positive initial energy; Energy estimate; Upper and lower bounds
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