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

In this article, we consider an energy-critical quintic wave equation on a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^3$ with nonlinear and nonlocal Kelvin–Voigt damping of the form $-\mathrm{\parallel }\mathrm{\nabla }{u}_t{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}^{\alpha }\mathrm{\Delta }{u}_t$, where $\alpha \in {\mathbb{R}}_{+}=[0,\mathrm{\infty })$. 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 $\mathcal{H}=H_0^1(\mathrm{\Omega })\times L^2(\mathrm{\Omega })$. 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 ${\mathcal{H}}_1=(H^2(\mathrm{\Omega })\cap H_0^1(\mathrm{\Omega }))\times H_0^1(\mathrm{\Omega })$. 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.