Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Stability of Single Transition Layer in Mass-Conserving Reaction-Diffusion Systems with Bistable Nonlinearity

Hideo Ikeda, Masataka Kuwamura

TL;DR

The paper studies mass-conserving reaction-diffusion systems with bistable nonlinearity and proves the existence of stationary single-transition-layer solutions using an analytic singular perturbation approach. Stability is characterized via the Evans function by computing the spectrum of the linearized operator around these layers; the key finding is that the sign of $J'(v^*)$ completely determines stability. Concretely, there is a unique small eigenvalue $\\\lambda(\\varepsilon) = O(\\varepsilon)$ whose real part has the same sign as $-J'(v^*)$, establishing stability when $J'(v^*)>0$ and instability when $J'(v^*)<0$. The work also clarifies the relation between the SLEP framework and the Evans-function analysis for mass-conserving systems, showing that the Evans function provides a necessary and sufficient stability criterion in this setting.

Abstract

Mass-conserving reaction-diffusion systems with bistable nonlinearity are considered under general assumptions. The existence of stationary solutions with a single internal transition layer in such reaction-diffusion systems is shown using the analytical singular perturbation theory. Moreover, a stability criterion for the stationary solutions is provided by calculating the Evans function.

Stability of Single Transition Layer in Mass-Conserving Reaction-Diffusion Systems with Bistable Nonlinearity

TL;DR

The paper studies mass-conserving reaction-diffusion systems with bistable nonlinearity and proves the existence of stationary single-transition-layer solutions using an analytic singular perturbation approach. Stability is characterized via the Evans function by computing the spectrum of the linearized operator around these layers; the key finding is that the sign of completely determines stability. Concretely, there is a unique small eigenvalue whose real part has the same sign as , establishing stability when and instability when . The work also clarifies the relation between the SLEP framework and the Evans-function analysis for mass-conserving systems, showing that the Evans function provides a necessary and sufficient stability criterion in this setting.

Abstract

Mass-conserving reaction-diffusion systems with bistable nonlinearity are considered under general assumptions. The existence of stationary solutions with a single internal transition layer in such reaction-diffusion systems is shown using the analytical singular perturbation theory. Moreover, a stability criterion for the stationary solutions is provided by calculating the Evans function.
Paper Structure (20 sections, 10 theorems, 251 equations, 1 figure)

This paper contains 20 sections, 10 theorems, 251 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Existence of single transition layer solutions
  3. Solutions of \ref{['b5']} and \ref{['b6']}
  4. $C^1$-matching at $x = x^*(\varepsilon)$
  5. Computation of \ref{['b1_1']}
  6. Determination of $\beta(\varepsilon)$ and $x^*(\varepsilon)$
  7. Stability analysis of the transition layer solutions
  8. Case $(\mathrm{I}) \ \lambda = \lambda(\varepsilon) = O(\varepsilon)$ as $\varepsilon \to 0$
  9. Construction of $\bar{V}_1$ and $\bar{V}_2$
  10. Construction of $\bar{V}_3$ and $\bar{V}_4$
  11. Evans function $g(\varepsilon;\varepsilon \kappa)$ corresponding to (\ref{['4c1']}) in the case (I)
  12. Case $(\mathrm{II}) \ \varepsilon \omega(\varepsilon) \to 0$ and $\omega(\varepsilon) \to \infty$ as $\varepsilon \to 0$
  13. Construction of $\bar{V}_1$ and $\bar{V}_2$
  14. Construction of $\bar{V}_3$ and $\bar{V}_4$
  15. Evans function $g(\varepsilon;\varepsilon \omega(\varepsilon)\hat{\lambda}(\varepsilon))$ corresponding to (\ref{['4c1']}) in the case (II)
  16. ...and 5 more sections

Key Result

Theorem 1.1

Under the assumptions (A1) - (A4), for any given $\xi$ satisfying a6, the mass-conserving reaction-diffusion system a1 has a family of single transition layer solutions $(u,v)(x;\varepsilon)$ satisfying b1_1 for sufficiently small $\varepsilon > 0$. Moreover, $(u,v)(x;\varepsilon)$ are stable if $J'

Figures (1)

  • Figure 1: Schematic profile of a single jump-up transition layer solution. This profile does not represent the $O(\varepsilon^2)$-amplitude transition layer of $v$-component because it is not required in our analysis.

Theorems & Definitions (15)

  • Theorem 1.1
  • Remark 1.1
  • Lemma 2.1
  • Proposition 2.1
  • Theorem 2.1
  • Remark 2.1
  • Remark 2.2
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • ...and 5 more