On the regularity for thermoelastic systems of phase-lag parabolic type

TL;DR

This paper studies the smoothness and regularity of solutions to thermoelastic systems of phase-lag type, proving analyticity for one model and Gevrey class membership for another, revealing strong stability properties.

Contribution

It establishes optimal regularity results for two thermoelastic models, showing analyticity and Gevrey class membership under specific conditions, which advances understanding of solution smoothness.

Findings

The semigroup for the fully thermoelastic plate model is analytic.

The semigroup for the partially thermoelastic plate model is in Gevrey class 4 for radial symmetric solutions.

Both models exhibit strong dissipative regularity properties.

Abstract

In this article, we investigate the maximal smoothness (infinite differentiability) of solutions to thermoelastic models, specifically those where the heat equation is of the ``phase-lag'' or ``parabolic'' type. We derive optimal regularity results for two distinct models. The first model addresses the transverse oscillations of a fully thermoelastic plate, for which we prove that the associated semigroup is analytic. The second model considers a partially thermoelastic plate composed of two components: a thermoelastic component with nonzero temperature differences and an elastic component unaffected by temperature variations. For this model, we demonstrate that the semigroup \( S(t) \) belongs to the Gevrey class of order 4, provided the solutions are radial and symmetric. Both analyticity and Gevrey class membership are qualitative properties that intricately link regularity and stability, driven by robust dissipative mechanisms. These properties are significantly stronger than standard regularity conditions, such as belonging to the class \( C^k \) or a Sobolev space \( H^s \).