Long-Time Behavior of Quasilinear Thermoelastic Kirchhoff-Love Plates with Second Sound

TL;DR

This paper analyzes the long-term behavior of a thermoelastic Kirchhoff-Love plate model with second sound, establishing global existence, uniqueness, and exponential decay of solutions under small initial data.

Contribution

It introduces a quasilinear hyperbolic system with second sound effects and proves global well-posedness and stabilization results for the model.

Findings

Global existence and uniqueness of solutions

Exponential decay of solutions over time

Stability under small initial data

Abstract

We consider an initial-boundary-value problem for a thermoelastic Kirchhoff & Love plate, thermally insulated and simply supported on the boundary, incorporating rotational inertia and a quasilinear hypoelastic response, while the heat effects are modeled using the hyperbolic Maxwell-Cattaneo-Vernotte law giving rise to a 'second sound' effect. We study the local well-posedness of the resulting quasilinear mixed-order hyperbolic system in a suitable solution class of smooth functions mapping into Sobolev $H^{k}$-spaces. Exploiting the sole source of energy dissipation entering the system through the hyperbolic heat flux moment, provided the initial data are small in a lower topology (basic energy level corresponding to weak solutions), we prove a nonlinear stabilizability estimate furnishing global existence & uniqueness and exponential decay of classical solutions.