Dynamics for a viscoelastic beam equation with past history and nonlocal boundary dissipation

TL;DR

This paper investigates the long-term behavior of a viscoelastic beam equation with past history and nonlocal boundary conditions, establishing the existence of a finite-dimensional global attractor and an exponential attractor under certain conditions.

Contribution

It extends previous models by considering infinite past history and proves the existence of compact and exponential attractors with finite dimension.

Findings

Existence of a compact global attractor in the weak phase space.

Finite dimensionality of the attractor in the subcritical case.

Existence of a fractal exponential attractor with finite dimension.

Abstract

This article aims to study the long-time dynamics of the linear viscoelastic plate equation $\displaystyle{u_{tt}+\Delta^2 u-\int_{\tau}^tg(t-s)\Delta^2u(s)ds=0}$ subject to nonlinear and nonlocal boundary conditions. This model, with $\tau=0$, was first considered by Cavalcanti (Discrete Contin. Dyn. Syst., 8(3), 675-695, 2002), where results of global existence and uniform decay rates of energy have been established. In this work, by taking $\tau=-\infty$, and considering the autonomous equivalent problem we prove that the dynamical system $(\mathcal{H},S_t)$ generated by the weak solutions has a compact global attractor $\mathfrak{A}$ (in the topology of the weak phase space $\mathcal{H}$), which in subcritical case has finite dimension and smoothness. Furthermore, when the force follows the {\it Hook Law}, we prove that $(\mathcal{H},S_t)$ possesses a (generalized) fractal exponential attractor $\mathfrak{A}_{\exp}$ with finite dimension in a space $\widetilde{\mathcal{H}}\supset\mathcal{H}$.