Compactness Properties and General Stability Result for an Abstract Nonlinear Viscoelastic Equation

This paper develops a unified method to derive compactness properties and convergence to a steady state as well as decay estimates of global bounded solutions of the abstract semilinear second order integro-differential equation u(tt) (t) + Au(t) - integral(tau)(0) k(s) Au(t - s)ds + f (u) = g(t), tau is an element of {t, infinity}, with finite (tau = t) or infinite (tau = infinity) memory. Here, A is a continuous linear operator, k is a positive nonincreasing function, f is a nonlinear function such that A + f is the gradient of a potential satisfying the Lojasiewicz-Simon inequality, and the time-dependent right-hand side g decays to 0 at infinity. The general results are applied to particular semilinear evolution viscoelastic problems at the end of the paper.