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 ε-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∈ L3tL18x. Under the further assumption f∈ C2(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.
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BibTeX
@misc{lasiecka2026global,
title = {{Global Well-Posedness and Asymptotic Behavior of Energy-Damped Subquintic Wave Equations}},
author = {Irena Lasiecka and Yanan Li and Vando Narciso},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.29778},
eprint = {2609.29778},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29778},
}
RIS
TY - UNPB
TI - Global Well-Posedness and Asymptotic Behavior of Energy-Damped Subquintic Wave Equations
AU - Irena Lasiecka
AU - Yanan Li
AU - Vando Narciso
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2609.29778
UR - https://arxiv.org/abs/2609.29778
AB - 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 ε-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∈ L^3_tL^18_x. Under the further assumption f∈ C^2(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.
ER -