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Orbital stability of monostable waves for reaction-diffusion systems

Louis Garénaux

Abstract

We study stability of monostable waves for reaction-diffusion systems. When the solution is initially close to a fast wave profile in optimal topology, we prove convergence to a shifted profile. The proof relies on explicit resolvent kernels estimates, allowing to handle weakly localized perturbations. It allows phase shift construction even when the translational eigenvalue is not associated to a zero of the Evans function. We further discuss distinction between Evans and Fourier eigenmodes when the marginal group velocity are directed towards the wave interface.

Orbital stability of monostable waves for reaction-diffusion systems

Abstract

We study stability of monostable waves for reaction-diffusion systems. When the solution is initially close to a fast wave profile in optimal topology, we prove convergence to a shifted profile. The proof relies on explicit resolvent kernels estimates, allowing to handle weakly localized perturbations. It allows phase shift construction even when the translational eigenvalue is not associated to a zero of the Evans function. We further discuss distinction between Evans and Fourier eigenmodes when the marginal group velocity are directed towards the wave interface.
Paper Structure (28 sections, 29 theorems, 194 equations)

This paper contains 28 sections, 29 theorems, 194 equations.

Table of Contents

  1. Introduction
  2. Litterature
  3. Orbital stability
  4. Known phase shift constructions
  5. Unknown phase shift constructions
  6. Main statement
  7. Discussion
  8. Future directions
  9. Matrix valued weights
  10. Logarithmic shifts
  11. Critical speed
  12. Orbital stability with respect to speed
  13. Notations
  14. Resolvent estimates
  15. Solutions of the eigenproblems
  16. ...and 13 more sections

Key Result

Theorem 1.3

Assume that Assumptions a:equilibrium, a:essential-spectrum and a:point-spectrum hold, and let $M\geq 1$. There exists positive constants $C$ and $\delta$ such that the following three points hold.

Theorems & Definitions (65)

  • Definition 1.1: Dispersion relations
  • Definition 1.2
  • Theorem 1.3
  • Lemma 1.4
  • proof
  • Remark 1.5
  • Remark 1.6
  • Lemma 2.1
  • proof
  • Remark 2.2
  • ...and 55 more