Global Attractor for Damped Wave Equations with Nonlinear Memory

Abstract:

      Let $\Omega\subset{\Bbb{R}}^{n}$ be a bounded domain with a smooth boundary. We consider the longtime dynamics of a class of damped wave equations with a nonlinear memory term $$u_{tt}+\alpha u_{t} -\Delta u - \int_{0}^{t}\mu (t-s)|u(s)|^{\beta} u(s)\d s + g(u)=f.$$ Based on a time-uniform priori estimate method, the existence of the compact global attractor is proved for this model in the phase space ${H_{0}^{1}(\Omega) \times L^{2}(\Omega)}$.