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

Abstract

This article aims to study the long-time dynamics of the linear viscoelastic plate equation $u_{tt}+{\Delta }^{2}u-{\int }_{\tau }^{t}g(t-s){\Delta }^{2}u(s)ds=0$ 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 $\tilde{\mathcal{H}}\supset \mathcal{H}$.