## Differential and Integral Equations

### 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.

#### Article information

Source
Differential Integral Equations, Volume 30, Number 11/12 (2017), 975-988.

Dates
First available in Project Euclid: 1 September 2017