Dynamics of strongly damped wave equations in locally uniform spaces: Attractors and asymptotic regularity

This paper is dedicated to analyzing the dynamical behavior of strongly damped wave equations with critical nonlinearity in locally uniform spaces. After proving the global well-posedness, we first establish the asymptotic regularity of the solutions which appears to be optimal and the existence of a bounded (in $H^2_{lu}(\mathbb{R}^N)\times H^1_{lu}(\mathbb{R}^N)$) subset which attracts exponentially every initial $H^1_{lu}(\mathbb{R}^N)\times L^2_{lu}(\mathbb{R}^N)$-bounded set with respect to the $H^1_{lu}(\mathbb{R}^N)\times L^2_{lu}(\mathbb{R}^N)$-norm. Then, we show there is a $(H ^1_{lu}(\mathbb{R}^N)\times L^2_{lu}(\mathbb{R}^N), H^1_\rho(\mathbb{R}^N)\times H^1_\rho(\mathbb{R}^N))$-global attractor, which reflects the strongly damped property of $\Delta u_t$ to some extent.