Stability of Global Bounded Solutions to a Nonautonomous Nonlinear Second Order Integro-Differential Equation

We study the long-time behavior as time goes to infinity of global bounded weak solutions to the following integro-differential equation $$\ddot u+k*\dot u+\nabla E(u)=g, $$ in finite dimensions, where the nonlinear potential $E$ satisfies the Łojasiewicz inequality near some equilibrium point. Based on an appropriate new Lyapunov function and Łojasiewicz inequality we prove that any global bounded weak solution converges to a steady state. We also obtain the rate of convergence according to the Łojasiewicz exponent and the time-dependent right-hand side $g$.