Attractors and Singular Limits for a Quintic Wave Equation with Nonlocal Kelvin--Voigt Damping

Abstract

In this article, we consider an energy-critical quintic wave equation on a bounded domain $\Omega \subset {\mathbb{R}}^3$ with nonlinear and nonlocal Kelvin--Voigt damping of the form $-\Vert \nabla u_t{\Vert }_{L^2(\Omega )}^{\alpha }\Delta u_t$, where $\alpha \in {\mathbb{R}}_{+}=[0,\infty )$. Under suitable hypotheses on the quintic source term, we establish the well-posedness of the problem and investigate its long-time dynamics in the natural energy space $\mathcal{H}=H_0^1(\Omega )\times L^2(\Omega )$. For every $\alpha \in {\mathbb{R}}_{+}$, we show that the associated dynamical system $(\mathcal{H},S^{\alpha }(t))$ is gradient and dissipative, and we prove a stabilization estimate that yields asymptotic smoothness and, consequently, the existence of a compact global attractor ${\mathcal{A}}_{\alpha }$. The same estimate provides an upper bound for the Kolmogorov $\epsilon$-entropy of ${\mathcal{A}}_{\alpha }$ and, in the limiting case $\alpha =0$, reduces to a quasi-stability inequality, which implies that ${\mathcal{A}}_0$ has finite fractal dimension. Furthermore, we prove that the family $\{{\mathcal{A}}_{\alpha }{\}}_{\alpha \in {\mathbb{R}}_{+}}$ is uniformly bounded in the higher-regularity space ${\mathcal{H}}_1=(H^2(\Omega )\cap H_0^1(\Omega ))\times H_0^1(\Omega )$. Finally, we establish the upper semicontinuity of $\{{\mathcal{A}}_{\alpha }{\}}_{\alpha \in {\mathbb{R}}_{+}}$ at $\alpha =0$, showing that the attractors associated with the nonlinear and nonlocal Kelvin--Voigt damping converge to the global attractor of the limiting problem with classical linear Kelvin--Voigt damping.

32 pages