Energy decay rates via convexity for some second-order evolution equation with memory and nonlinear time-dependent dissipation

The stabilization of the following abstract integro-differential equation: u″(t)+Au(t)+∫0tg(t−s)Au(s)ds+Q(t,u′(t))=∇F(u(t)), is investigated. We establish the general decay rate of the solution energy in terms of the behavior of the nonlinear feedback and the relaxation function gg, without imposing any restrictive growth assumption on the damping at the origin and strongly weakening the usual assumption of the relaxation function gg. Our approach is based on the multiplier method and make use of some properties of the convex functions. These decay results can be applied to various concrete models. We shall study some examples to illustrate our result.