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Convergence to equilibrium for a degenerate three species reaction-diffusion system

Saumyajit Das, Harsha Hutridurga

Abstract

In this work, we study a $3\times 3$ triangular reaction-diffusion system. Our main objective is to understand the long time behaviour of solutions to this reaction-diffusion system when there are degeneracies. More precisely, we treat cases when one of the diffusion coefficients vanishes while the other two diffusion coefficients stay positive. We prove convergence to equilibrium type results. In all our results, the constants appearing in the decay estimates are explicit.

Convergence to equilibrium for a degenerate three species reaction-diffusion system

Abstract

In this work, we study a triangular reaction-diffusion system. Our main objective is to understand the long time behaviour of solutions to this reaction-diffusion system when there are degeneracies. More precisely, we treat cases when one of the diffusion coefficients vanishes while the other two diffusion coefficients stay positive. We prove convergence to equilibrium type results. In all our results, the constants appearing in the decay estimates are explicit.

Paper Structure

This paper contains 4 sections, 19 theorems, 301 equations.

Table of Contents

  1. Introduction
  2. The case of $d_b=0$
  3. The case of $d_c=0$
  4. some useful results

Key Result

Theorem 1

For $N\geq4$, let $(a,b,c)$ be the solution to the degenerate system eq:model 1. Let $(a_\infty,b_\infty,c_\infty)$ be the associated equilibrium state given by eq:equi-state-1-eq:equi-state-2. Let the nonzero diffusion coefficients $d_a,d_c$ satisfy the closeness condition closeness condition imp. For $N<4$, let $(a,b,c)$ be the solution to the degenerate system eq:model 1. Let $(a_\infty,b_\inf

Theorems & Definitions (32)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Proposition 3
  • proof
  • Lemma 3
  • proof
  • ...and 22 more