Existence of time-dependent attractor of wave equation in locally uniform space

Abstract

In this article, we study non-autonomous dynamical behavior of weakly damped wave equation in unbounded domain. First of all, we introduce the time-dependent locally uniform space. After that, the pullback asymptotical compactness is proved by applying the contractive function method. Eventually, we obtain the existence of $(H_{\mathrm{lu}}^1({\mathbb{R}}^3)\times L_{\mathrm{lu}}^2({\mathbb{R}}^3),H_{\rho }^1({\mathbb{R}}^3)\times L_{\rho }^1({\mathbb{R}}^3))$ -time-dependent attractor of wave equation.