Exponential decay for waves with indefinite memory dissipation

Abstract

In this work, we deal with the following wave equation with localized dissipation given by a memory term $$ u_{tt} -u_{xx} + \partial_x \Big\{ a(x)\int_{0}^{t} g(t-s)u_{x}(x,s)ds \Big\}=0. $$ We consider that this dissipation is indefinite due to sign changes of the coefficient $a$ or by sign changes of the memory kernel $g$. The exponential decay of solutions is proved when the average of coefficient $a$ is positive and the memory kernel $g$ is small.