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

TL;DR

This paper proves the existence and uniqueness of solutions for a beam equation with degenerate nonlocal damping and establishes the existence of a strong attractor in a higher topology space.

Contribution

It addresses an open question by proving well-posedness and constructs a strong attractor for the beam model with nonlocal damping.

Findings

Global existence and uniqueness of weak solutions

Existence of a strong attractor in a higher topology space

Positive answer to an open problem in the literature

Abstract

This paper is devoted to initial-boundary value problem of an extensible beam equation with degenerate nonlocal energy damping in $\Omega\subset\mathbb{R}^n$: $u_{tt}-\kappa\Delta u+\Delta^2u-\gamma(\Vert \Delta u\Vert^2+\Vert u_t\Vert^2)^q\Delta u_t+f(u)=0$. 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 $\mathcal{H}_{\frac{1}{q}}$.