Local well-posedness and blow-up of solution for a higher-order wave equation with viscoelastic term and variable-exponent

Document Type : Research Paper


1 Department of Mathematics, Laboratory of Applied Mathematics and History and Didactics of Mathematics (LAMAHIS), University of 20 August 1955, Skikda, Algeria

2 Department of Sciences and Technology, Laboratory of Applied Mathematics and History and Didactics of Mathematics (LAMAHIS), University of 20 August 1955, Skikda, Algeria


We investigate in this paper a value problem related to the following nonlinear higher-order wave equation $$
    \eta_{tt}+\left(  -\Delta\right)  ^{m}\eta-%
    %TCIMACRO{\dint \limits_{0}^{t}}%
    g\left(  t-s\right)  \left(  -\Delta\right)  ^{m}\eta\left(  s\right)
    ds+\eta_{t}=\left\vert \eta\right\vert ^{p\left(  x\right)  -2}\eta.
Firstly, we prove the existence and uniqueness of the local solution under suitable conditions for the relaxation function $g$ and viable-exponent $p\left(  .\right)  $, using a method, which is a mixture of the Faedo-Galarkin and Banach fixed point theorem, and prove also the solution blows up in finite time. Finally, we give a two-dimensional numerical example to illustrate the blow-up result.


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Volume 14, Issue 4
April 2023
Pages 111-124
  • Receive Date: 23 December 2022
  • Revise Date: 15 February 2023
  • Accept Date: 15 February 2023