Exponential energy decay of solutions of viscoelastic wave equations

In this paper we consider the nonlinear viscoelastic equation utt − u + t 0 g(t −τ) u(τ )dτ + ut = |u|p−1u in Ω ×(0,∞), with initial conditions and Dirichlet boundary condition. Under some appropriate assumption on g, by introducing a potential well we establish the global existence result for the equation above. Furthermore, we investigate explicit exponential energy decay of the viscoelastic wave equation under the potential well