Long-time Behavior for a Nonlinear Plate Equation with Thermal Memory

TL;DR

This paper studies the long-term behavior of a nonlinear plate equation with thermal memory, proving existence, uniqueness, and convergence of solutions to steady states, along with convergence rates.

Contribution

It introduces a novel analysis of a nonlinear plate equation with thermal memory, establishing global solutions, attractors, and convergence rates using a Lojasiewicz--Simon inequality.

Findings

Existence and uniqueness of global solutions

Existence of a global attractor

Solutions converge to steady states with estimated rates

Abstract

We consider a nonlinear plate equation with thermal memory effects due to non-Fourier heat flux laws. First we prove the existence and uniqueness of global solutions as well as the existence of a global attractor. Then we use a suitable Lojasiewicz--Simon type inequality to show the convergence of global solutions to single steady states as time goes to infinity under the assumption that the nonlinear term $f$ is real analytic. Moreover, we provide an estimate on the convergence rate.