Pullback Attractor for a Weakly Damped Wave Equation with Sup-Cubic Nonlinearity

In this paper, the non-autonomous dynamical behavior of weakly damped wave equation with a sup-cubic nonlinearity is considered in locally uniform spaces. We first prove the global well-posedness of the Shatah-Struwe solutions, then establish 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}-pullback attractor for the Shatah-Struwe solutions process of this equation. The results are based on the recent extension of Strichartz estimates for the bounded domains.