General Decay of Solutions in a Viscoelastic Equation with Nonlinear Localized Damping

In this paper,we consider the following viscoelastic equation u tt- △u +∫t 0 g(t-s)△u(s)ds + a(x)u t + u |u|r = 0 with initial condition and Dirichlet boundary condition.The decay property of the energy function closely depends on the properties of the relaxation function g(t) at infinity.In the previous works of [3,7,11],it was required that the relaxation function g(t) decay exponentially or polynomially as t → +∞.In the recent work of Messaoudi [12,13],it was shown that the energy decays at a similar rate of decay of the relaxation function,which is not necessarily dacaying in a polynomial or exponential fashion.Motivated by [12,13],under some assumptions on g(x),a(x) and r,and by introducing a new perturbed energy,we also prove the similar results for the above equation.