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

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.