Dynamics of Fractional Wave Equations with Nonlocal Damping

Abstract

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.