AI 中文总结
本文研究带非局部Kelvin–Voigt阻尼的能量临界五次波动方程，建立其适定性，证明动力系统的耗散性、吸引子的存在性等，还分析吸引子族的一致有界性与上半连续性。
AI 中文摘要
本文研究三维有界区域Ω⊂ℝ³上的能量临界五次波动方程，其带有形式为−‖∇u_t‖_{L²(Ω)}^αΔu_t的非线性非局部Kelvin–Voigt阻尼，其中α∈ℝ₊=[0,∞)。在五次源项的适当假设下，我们建立了该问题的适定性，并在自然能量空间ℋ=H₀¹(Ω)×L²(Ω)中研究其长时间动力学。对每个α∈ℝ₊，我们证明对应的动力系统(ℋ,S^α(t))是梯度耗散的，且得到一个稳定化估计，该估计可推出渐近光滑性，进而得到紧整体吸引子𝒜_α的存在性。该估计还为𝒜_α的Kolmogorov ε-熵提供了上界，且在极限情形α=0时，该估计退化为拟稳定性不等式，这意味着𝒜_₀具有有限分形维数。此外，我们证明族{𝒜_α}_{α∈ℝ₊}在更高正则性空间ℋ₁=(H²(Ω)∩H₀¹(Ω))×H₀¹(Ω)中是一致有界的。最后，我们建立{𝒜_α}_{α∈ℝ₊}在α=0处的上半连续性，表明带有非线性非局部Kelvin–Voigt阻尼的吸引子收敛到具有经典线性Kelvin–Voigt阻尼的极限问题的整体吸引子。
英文摘要
In this article, we consider an energy-critical quintic wave equation on a bounded domain $Ω\subset\mathbb{R}^3$ with nonlinear and nonlocal Kelvin--Voigt damping of the form $-\|\nabla u_t\|_{L^2(Ω)}^αΔu_t$, where $α\in\mathbb{R}_+=[0,\infty)$. 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 $\mathcal H=H_0^1(Ω)\times L^2(Ω)$. For every $α\in\mathbb{R}_+$, we show that the associated dynamical system $(\mathcal H,S^α(t))$ is gradient and dissipative, and we prove a stabilization estimate that yields asymptotic smoothness and, consequently, the existence of a compact global attractor $\mathcal A_α$. The same estimate provides an upper bound for the Kolmogorov $\varepsilon$-entropy of $\mathcal A_α$ and, in the limiting case $α=0$, reduces to a quasi-stability inequality, which implies that $\mathcal A_0$ has finite fractal dimension. Furthermore, we prove that the family $\{\mathcal A_α\}_{α\in\mathbb{R}_+}$ is uniformly bounded in the higher-regularity space $\mathcal H_1=(H^2(Ω)\cap H_0^1(Ω))\times H_0^1(Ω)$. Finally, we establish the upper semicontinuity of $\{\mathcal A_α\}_{α\in\mathbb{R}_+}$ at $α=0$, 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.
Comments32 pages
