Energy decay rates of solutions to a viscoelastic wave equation with variable exponents and weak damping

TL;DR

This paper investigates the long-term energy decay of solutions to a viscoelastic wave equation with variable exponents and weak damping, deriving general decay results and specific polynomial and exponential decay rates.

Contribution

It extends previous studies by considering variable exponents and weak damping, providing new decay rate results for the viscoelastic wave equation.

Findings

General decay results under specific conditions on g(t)

Exponential decay when g decays polynomially

Polynomial decay rates for certain g(t) functions

Abstract

The goal of the present paper is to study the asymptotic behavior of solutions for the viscoelastic wave equation with variable exponents \[ u_{tt}-\Delta u+\int_0^tg(t-s)\Delta u(s)ds+a|u_t|^{m(x)-2}u_t=b|u|^{p(x)-2}u\] under initial-boundary condition, where the exponents $p(x)$ and $m(x)$ are given functions, and $a,~b>0$ are constants. More precisely, under the condition $g'(t)\le -\xi(t)g(t)$, here $\xi(t):\mathbb{R}^+\to\mathbb{R}^+$ is a non-increasing differential function with $\xi(0)>0,~\int_0^\infty\xi(s)ds=+\infty$, general decay results are derived. In addition, when $g$ decays polynomially, the exponential and polynomial decay rates are obtained as well, respectively. This work generalizes and improves earlier results in the literature.