Analysis of the Energy Decay of a Degenerated Thermoelasticity System

TL;DR

This paper investigates how energy decays in a thermoelastic system with a degenerated heat operator, revealing different propagation behaviors depending on the degeneracy set.

Contribution

It provides a detailed analysis of energy evolution in degenerated thermoelastic systems, including cases with different degeneracy geometries.

Findings

Energy density follows damped wave propagation in certain degenerate cases.

Propagation of energy is distorted along rays when degeneracy occurs in an open set.

The study extends understanding of energy decay in systems with degenerated operators.

Abstract

In this paper, we study a system of thermoelasticity with a degenerated second order operator in the Heat equation. We analyze the evolution of the energy density of a family of solutions. We consider two cases: when the set of points where the ellipticity of the Heat operator fails is included in a hypersurface and when it is an open set. In the first case and under special assumptions, we prove that the evolution of the energy density is the one of a damped wave equation: propagation along the rays of geometric optic and damping according to a microlocal process. In the second case, we show that the energy density propagates along rays which are distortions of the rays of geometric optic.