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Global Nonexistence for A Viscoelastic Wave Equation with Acoustic Boundary Conditions

Authors
  • Yu, Jiali1, 2
  • Shang, Yadong2
  • Di, Huafei2
  • 1 Dalian Jiaotong University, Dalian, 116028, China , Dalian (China)
  • 2 Guangzhou University, Guangzhou, 510006, China , Guangzhou (China)
Type
Published Article
Journal
Acta Mathematica Scientia
Publisher
Springer-Verlag
Publication Date
Dec 17, 2019
Volume
40
Issue
1
Pages
155–169
Identifiers
DOI: 10.1007/s10473-020-0111-2
Source
Springer Nature
Keywords
License
Yellow

Abstract

This paper deals with a class of nonlinear viscoelastic wave equation with damping and source terms \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${u_{tt}} - \Delta u - \Delta {u_t} - \Delta {u_{tt}} + \int_0^t {g(t - s)\Delta u(s)ds + {u_t}{u_t}{^{m - 2}} = u|u|{^{p - 2}}} $\end{document} with acoustic boundary conditions. Under some appropriate assumption on relaxation function g and the initial data, we prove that the solution blows up in finite time if the positive initial energy satisfies a suitable condition.

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