WELL-POSEDNESS AND ASYMPTOTIC BEHAVIOR OF A NONAUTONOMOUS, SEMILINEAR HYPERBOLICPARABOLIC EQUATION WITH DYNAMICAL BOUNDARY CONDITION OF MEMORY TYPE

摘要

We consider a nonautonomous, semilinear, hyperbolic-parabolic equation subject to a dynamical boundary condition of memory type. First we prove the existence and uniqueness of global bounded solutions having relatively compact range in the natural energy space. Under the assumption that the nonlinear term f is real analytic, we then derive an appropriate Lyapunov energy and we use the Lojasiewicz-Simon inequality to show the convergence of global weak solutions to single steady states as time tends to infinity. Finally, we provide an estimate for the convergence rate.