Energy decay of wave equations with infinite memory effects versus supercritical frictional dampings

Abstract

In this paper, a class of damped viscoelastic wave equations

$$\begin{aligned} u_{tt}-k(0)\Delta u-\int _0^\infty k'(s)\Delta u(t-s)ds+|u_t|^{m-1}u_t=|u|^{p-1}u \end{aligned}$$

is considered in a bounded domain \(\Omega \subset {\mathbb {R}}^3\). Uniform energy decay was discussed which depends on the relaxation function \(-k'(s)\) in the previous work (Guo et al., Z Angew Math Phys 69:65, 2018) for \(1\le m\le 5\). Depending on a key integral inequality obtained by Martinez (ESAIM Control Optim Calc Var 4:419–444, 1999), we establish the decay estimate of the total energy for \(m>5\). Our results improve and complement the previous one. As an example, a logarithmic energy decay is also presented.