Dynamics for a viscoelastic beam equation with past history and nonlocal boundary dissipation
Linfang Liu, Vando Narciso, Zhijian Yang
TL;DR
This work analyzes the long-time dynamics of a viscoelastic beam/plate equation with memory and nonlinear, nonlocal boundary damping. By reformulating with past-history variable $\eta^t$ and working in an autonomous framework, the authors prove the existence and uniqueness of global solutions and build a dissipative gradient dynamical system on a suitable phase space $\mathcal{H}$. They establish a compact global attractor $\mathfrak{A}$ (finite fractal dimension for subcritical $p<p^*$) and, under Hook Law, a generalized fractal exponential attractor $\mathfrak{A}_{\exp}$ in an extended space $\widetilde{\mathcal{H}}$, with Hölder continuity results enabling exponential attraction. The analysis leverages energy methods, a quasi-stability framework, and an auxiliary history variable to control memory effects, improving previous results and providing a detailed description of asymptotic behavior for this class of viscoelastic systems.
Abstract
This article aims to study the long-time dynamics of the linear viscoelastic plate equation $\displaystyle{u_{tt}+Δ^2 u-\int_τ^tg(t-s)Δ^2u(s)ds=0}$ subject to nonlinear and nonlocal boundary conditions. This model, with $τ=0$, 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 $τ=-\infty$, 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 $\widetilde{\mathcal{H}}\supset\mathcal{H}$.