Long-time dynamics of the wave equation with nonlocal weak damping and sup-cubic nonlinearity in 3-D domains

In this paper, we study the long-time dynamics for the wave equation with nonlocal weak damping and sup-cubic nonlinearity in a bounded smooth domain of ${\mathbb{R}}^3\mathrm{.}$ Based on the Strichartz estimates for the case of bounded domains, we first prove the global well-posedness of the Shatah-Struwe solutions. Then we establish the existence of the global attractor for the Shatah-Struwe solution semigroup by the method of contractive function. Finally, we verify the existence of a polynomial attractor for this semigroup.