Existence and Asymptotic Behavior of Solutions to Semilinear Wave Equations with Nonlinear Damping and Dynamical Boundary Condition

Abstract

Our aim in this article is to study a nonautonomous semilinear wave equation with nonlinear damping and dynamical boundary condition. First we prove the existence and uniqueness of global bounded solutions having relatively compact range in the natural energy space. Then, by deriving an appropriate Lyapunov energy, we show that if the exponent in the Łojasiewicz-Simon inequality is large enough (depending on the damping), then weak solutions converge to equilibrium.