International Journal of Nonlinear Analysis and Applications
Local well-posedness and blow-up of solution for a higher-order wave equation with viscoelastic term and variable-exponent
Document Type : Research Paper
Authors
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
Abstract
We investigate in this paper a value problem related to the following nonlinear higher-order wave equation $$
\eta_{tt}+\left( -\Delta\right) ^{m}\eta-%
%TCIMACRO{\dint \limits_{0}^{t}}%
%BeginExpansion
{\displaystyle\int\limits_{0}^{t}}
%EndExpansion
g\left( t-s\right) \left( -\Delta\right) ^{m}\eta\left( s\right)
ds+\eta_{t}=\left\vert \eta\right\vert ^{p\left( x\right) -2}\eta.
$$
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.
Keywords
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Volume 14, Issue 4
April 2023Pages 111-124
- Receive Date: 23 December 2022
- Revise Date: 15 February 2023
- Accept Date: 15 February 2023