Blow up for initial-boundary value problem of wave equation with a nonlinear memory in 1-D

Abstract

The present paper is devoted to studying the initial-boundary value problem of a 1-D wave equation with a nonlinear memory: ${u_{tt}} - {u_{xx}} = \frac{1}{{\Gamma \left( {1 - \gamma } \right)}}\int_0^t {{{\left( {t - s} \right)}^{ - \gamma }}{{\left| {u\left( s \right)} \right|}^p}ds} .$ The blow up result will be established when p > 1 and 0 < γ < 1, no matter how small the initial data are, by introducing two test functions and a new functional.

Keywords

Blow up / Wave equation / Nonlinear memory / Initial-boundary value problem

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