Thermoelastic plates with type I heat conduction with second gradient

TL;DR

This paper studies the stability and behavior of thermoelastic plates modeled with second-gradient theory and Type I heat conduction, revealing stability, non-differentiability, and instability phenomena under various conditions.

Contribution

It proves the exponential stability of solutions, analyzes non-differentiability of the semigroup, and examines solution behavior with negative elastic parameters and in one-dimensional quasi-static cases.

Findings

Semigroup is non-differentiable with bi-Laplacian in heat equation

Solutions are unstable when elastic parameter is negative

Existence and exponential decay of solutions in 1D quasi-static case

Abstract

This paper investigates the qualitative properties of thermoelastic plates modeled by the second-gradient theory with a Type I heat equation. We establish the exponential stability of the solutions. Our main contribution is to prove that the semigroup is non-differentiable when the bi-Laplacian operator appears in the heat equation. Additionally, we analyze the case where the elastic parameter is negative, demonstrating the uniqueness and instability of the solutions. Finally, in the one-dimensional quasi-static case, we demonstrate the existence and exponential decay of the solutions under specific conditions.