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

Abstract

In this article, we consider a wave equation on a bounded three-dimensional domain, featuring a degenerate nonlocal damping mechanism that depends on the system's energy and a source term with subquintic growth. This class of models is motivated by applications in optical physics. We establish global existence in the Shatah--Strichartz sense by combining Galerkin approximations with space-time Strichartz estimates on bounded domains. A central difficulty in the long-time analysis is the interplay between three-dimensional wave dynamics, nonlocal damping, and forcing terms that are not locally Lipschitz continuous. We prove the existence of a compact global attractor and derive an upper bound for its Kolmogorov $\epsilon$-entropy. In the non-degenerate damping case, assuming that the domain has a smooth, strictly convex boundary, we exploit the refined Strichartz estimates of Ivanovici {\it et al.}~\cite{Ivanovici-Lascar-Lebeau-Planchon-2023}, which yield, in particular, $u\in L_t^3{L}_x^{18}$. Under the further assumption $f\in C^2(\mathbb{R})$, this improved integrability allows us to establish quasi-stability of the associated dynamical system. Consequently, the global attractor has finite fractal dimension and higher regularity, and a generalized exponential attractor exists.

arXiv comment42 pages