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Stabilization for degenerate equations with drift and small singular term

Genni Fragnelli, Dimitri Mugnai, Amine Sbai

Abstract

We consider a degenerate/singular wave equation in one dimension, with drift and in presence of a leading operator which is not in divergence form. We impose a homogeneous Dirichlet boundary condition where the degeneracy occurs and a boundary damping at the other endpoint. We provide some conditions for the uniform exponential decay of solutions for the associated Cauchy problem.

Stabilization for degenerate equations with drift and small singular term

Abstract

We consider a degenerate/singular wave equation in one dimension, with drift and in presence of a leading operator which is not in divergence form. We impose a homogeneous Dirichlet boundary condition where the degeneracy occurs and a boundary damping at the other endpoint. We provide some conditions for the uniform exponential decay of solutions for the associated Cauchy problem.
Paper Structure (6 sections, 13 theorems, 150 equations)

This paper contains 6 sections, 13 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Preliminary results and well posedness
  3. Stability result
  4. Appendix
  5. Proof of Theorem \ref{['thmmildclassolution']}
  6. Aknowledgments

Key Result

Proposition 2.1

Assume Hypothesis hyp1. Then, there exists $C>0$ such that for all $u \in H^1_{\frac{1}{\sigma},0}(0,1)$, Moreover, if $K_1+ K_2 \le 2$ and $u \in H^1_{\frac{1}{\sigma},0}(0,1)$ then $\frac{u}{\sqrt{\sigma d}} \in L^2(0,1)$ and there exists a positive constant $C>0$ such that

Theorems & Definitions (27)

  • Definition 1.1
  • Definition 1.2
  • Remark 1
  • Remark 2
  • Proposition 2.1
  • Corollary 2.1
  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.1
  • Proposition 2.2
  • ...and 17 more