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The goal of the present paper is to study the asymptotic behavior ofsolutions 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 $m(x)$ aregiven functions, and $a,~b>0$ are constants. More precisely, under thecondition $g'(t)\le -\xi(t)g(t)$, here $\xi(t):\mathbb{R}^+\to\mathbb{R}^+$ isa 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 decayrates are obtained as well, respectively. This work generalizes and improvesearlier results in the literature.

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