Well-posedness and strong attractors for a beam model with degenerate nonlocal strong damping

Abstract

This paper is devoted to initial-boundary value problem of an extensible beam equation with degenerate nonlocal energy damping in $\Omega \subset R^n$: $u_{tt}-\kappa \Delta u+{\Delta }^{2}u-\gamma {({\Vert \Delta u\Vert }^2+{\Vert u_t\Vert }^2)}^q\Delta u_t+f\left(u\right)=0$. We prove the global existence, regularity and uniqueness of weak solutions. Moreover, we establish the existence of a strong attractor for the corresponding weak solution semigroup, where the “strong” means that the compactness and attractiveness of the attractor are in the topology of a stronger space $H_{\frac{1}{q}}$.