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Stability of a degenerate thermoelastic equation

Kaïs Ammari, Fathi Hassine, Luc Robbiano

Abstract

This work is dedicated to the study of a linear model arising in thermoelastic rod of homogeneous material. The system is resulting from a coupling of a heat and a wave equation in the interval $(0,1)$ with Dirichlet boundary conditions at the outer endpoints where the parabolic component is degenerating at the end point $x=0$. Two models are considered the first is with weak degeneracy and the second is with strong degeneracy. We aim to study the well-posedness and asymptotic stability of both systems using techniques from the $C_{0}$-semigroup theory and a use a frequency domain approach based on the well-known result of Prüss in order to prove using some multiplier techniques that the energy of classical solutions decays uniformly as time goes to infinity.

Stability of a degenerate thermoelastic equation

Abstract

This work is dedicated to the study of a linear model arising in thermoelastic rod of homogeneous material. The system is resulting from a coupling of a heat and a wave equation in the interval with Dirichlet boundary conditions at the outer endpoints where the parabolic component is degenerating at the end point . Two models are considered the first is with weak degeneracy and the second is with strong degeneracy. We aim to study the well-posedness and asymptotic stability of both systems using techniques from the -semigroup theory and a use a frequency domain approach based on the well-known result of Prüss in order to prove using some multiplier techniques that the energy of classical solutions decays uniformly as time goes to infinity.

Paper Structure

This paper contains 5 sections, 14 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Well-posedness
  3. Stability result
  4. Strong stability
  5. Uniform decay rate

Key Result

Lemma 2.1

Let $\theta$ be such that $(a\theta)\in H_{\ell}^{1}(0,1) := \left\{u \in H^1(0,1), \, u(0) =0 \right\},$ then and if $a\theta'\in H^{1}(0,1)$ such that $(a\theta')(0)=0$,

Theorems & Definitions (26)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.1
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • ...and 16 more