Convergence to equilibrium for second order differential equations with weak damping of memory type

Abstract

We study the asymptotic behavior, as $t\to\infty$, of bounded solutions to a second-order integro-differential equation in finite dimensions where the damping term is of memory type and can be of arbitrary fractional order less than 1. We derive appropriate Lyapunov functions for this equation and prove that any global bounded solution converges to an equilibrium of a related equation, if the nonlinear potential ${\mathcal E}$ occurring in the equation satisfies the Łojasiewicz inequality.