带非局部阻尼的分数阶波动方程的动力学
Dynamics of Fractional Wave Equations with Nonlocal Damping
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中文总结 AI 辅助
本文研究带Balakrishnan–Taylor型非局部阻尼的分数阶波动方程，利用线性算子理论证明其温和解与正则解的整体适定性，刻画了相关半群的渐近动力学并建立全局吸引子，是该类方程渐近动力学的首次系统研究。
中文摘要 AI 辅助
本工作研究在有界域Ω⊂ℝⁿ上的具有Balakrishnan–Taylor型阻尼的分数阶波动方程。该模型将分数阶拉普拉斯算子与依赖于分数阶能量的非线性阻尼系数耦合，形成了一个双重非局部演化方程，推广了经典的具有能量相关耗散的波动方程。通过线性算子理论，我们建立了温和解和正则解的整体适定性。随后，我们研究了相关半群的长时间动力学，证明其是梯度型且渐近光滑的。由此，我们建立了紧全局吸引子的存在性，并表明该吸引子与定常解集合的不稳定流形重合。据我们所知，这是首次对具有Balakrishnan–Taylor型阻尼的分数阶波动方程的渐近动力学进行刻画。
英文摘要
In this work we study a fractional wave equation with Balakrishnan--Taylor type damping posed on a bounded domain $Ω\subset\mathbb{R}^n$. The model couples the fractional Laplacian with a nonlinear damping coefficient depending on the fractional energy, leading to a doubly nonlocal evolution equation that extends the classical wave equation with energy-dependent dissipation. By means of the theory of linear operators, we establish the global well-posedness of both mild and regular solutions. We then investigate the long-time dynamics of the associated semigroup and prove that it is gradient and asymptotically smooth. As a consequence, we establish the existence of a compact global attractor and show that it coincides with the unstable manifold of the set of stationary solutions. To the best of our knowledge, this provides the first characterization of the asymptotic dynamics for fractional wave equations with Balakrishnan--Taylor type damping.
