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 $\Omega \subset {\mathbb{R}}^n$: $u_{tt}-\kappa \Delta u+{\Delta }^{2}u-\gamma (\Vert \Delta u{\Vert }^2+\Vert u_t{\Vert }^2{)}^q\Delta u_t+f(u)=0$. We prove the global existence and uniqueness of weak solutions, which gives a positive answer to an open question in [24]. 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 ${\mathcal{H}}_{\frac{1}{q}}$.

27 pages