Global Well-Posedness and Asymptotic Behavior of Energy-Damped Subquintic Wave Equations

Abstract: In this paper, we consider a wave equation in a bounded three-dimensional domain with a degenerate nonlocal damping mechanism depending on the system's energy and a source term with subquintic growth. This kind of model is motivated by applications to optical physics. We establish global existence in the Shatah--Struwe sense by combining Galerkin approximations with space-time Strichartz estimates on bounded domains. Our main contributions concern long-time dynamics, which are particularly challenging for three-dimensional waves with forcing terms without local Lipschitz regularity. Specifically, we establish the existence of a compact global attractor and derive an upper bound for its Kolmogorov $\varepsilon$-entropy. Furthermore, when the damping coefficient is non-degenerate, we employ a novel combination of admissible Strichartz pairs, adapted to the regularity of the forcing term. This approach allows us to establish quasi-stability and, consequently, the finite dimensionality and higher regularity of the global attractor, as well as the existence of a generalized exponential attractor.