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 $\Omega\subset\mathbb{R}^3$ with nonlinear and nonlocal Kelvin--Voigt damping of the form $-\|\nabla u_t\|_{L^2(\Omega)}^\alpha\Delta u_t$, 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 $\varepsilon$-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.