Attractors for A sup-cubic weakly damped wave equation in $ \mathbb{R}^{3} $

In this paper, the dynamical behavior of weakly damped wave equations with a sup-cubic nonlinearity is considered in locally uniform spaces. We first prove the global well-posedness of the Shatah-Struwe solutions, then we obtain the existence of the \begin{document}$ \big(H_{lu}^{1}(\mathbb{R}^{3})\times L_{lu}^{2}(\mathbb{R}^{3}),H_{\rho}^{1}(\mathbb{R}^{3})\times L_{\rho}^{2}(\mathbb{R}^{3})\big) $\end{document} -global attractor for the Shatah-Struwe solutions semigroup of this equation. The results are crucially based on the recent extension of Strichartz estimates to the case of bounded domains.