Exponential Decay for a Nonlinear Viscoelastic Equation with Singular Kernels

The nonlinear viscoelastic wave equation |ut|ρutt-Δu-Δutt+∫0tg(t-s)Δu(s)ds+|u|pu=0, in a bounded domain with initial conditions and Dirichlet boundary conditions is considered. We prove that, for a class of kernels g which is singular at zero, the exponential decay rate of the solution energy. The result is obtained by introducing an appropriate Lyapounov functional and using energy method similar to the work of Tatar in 2009. This work improves earlier results.