Upper-semicontinuity of uniform attractors for the non-autonomous viscoelastic Kirchhoff plate equation with memory

TL;DR

This paper investigates the long-term behavior of a non-autonomous viscoelastic Kirchhoff plate equation with memory, establishing the existence and upper semicontinuity of uniform attractors as parameters vary, extending classical attractor theory.

Contribution

It introduces a novel analytical approach to prove the upper semicontinuity of uniform attractors in viscoelastic systems with memory and critical nonlinearities.

Findings

Existence of a global weak solution inducing a continuous process.

Existence of uniform attractors in subcritical and critical growth cases.

Proved upper semicontinuity of attractors as perturbation parameter tends to zero.

Abstract

This paper delves into the long-time dynamics of a non-autonomous viscoelastic Kirchhoff plate equation with memory effects, described by $$ u_{t t}-\Delta u_{t t}+a_\epsilon(t) u_t+\alpha \Delta^2 u-\int_0^{\infty} \mu(s) \Delta^2 u(t-s) \mathrm{d} s-\Delta u_t+f(u)=g(x,t), $$ in bounded domain $\Omega \subset \mathbb{R}^N$ with smooth boundary and nonlinear terms. Initially, the global existence of a weak solution that induces a continuous process is established. Subsequently, the existence of a uniform attractor is demonstrated in both subcritical and critical growth scenarios, utilizing operator techniques and an innovative analytical approach. Finally, the upper semicontinuity of the family of uniform attractors as the pert parameterurbation $\epsilon \to 0^+$ is proven through delicate energy estimates and a contradiction argument. Our results not only extend classical attractor theory to more general non-autonomous viscoelastic systems but also resolve open questions regarding the limiting behavior of attractors in the presence of both memory and critical nonlinearity.