Exponential Attractors for the Sup-Cubic Wave Equation with Nonlocal Damping

We study the long-time dynamics of a wave equation with nonlocal weak damping, nonlocal weak anti-damping and sup-cubic nonlinearity. Based on the Strichartz estimates in a bounded domain, we obtain the global well-posedness of the Shatah-Struwe solutions. To overcome the difficulties brought by the nonlinear weak damping term, we present a new-type Gronwall's lemma to obtain the dissipative for the Shatah-Struwe solutions semigroup of this equation. Finally, we establish the existence of a time-dependent exponential attractor with the help of a more general criteria constructed by the quasi-stable technique.