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Uniqueness and stability of monostable pulsating fronts for multi-dimensional reaction-diffusion-advection systems in periodic media

Li-Jun Du, Wan-Tong Li, Ming-Zhen Xin

TL;DR

The work extends the theory of pulsating traveling fronts to multi-component reaction-diffusion-advection systems in space-periodic media, proving uniqueness up to translation and global stability of fronts with nonzero speed. By analyzing exact front asymptotics in both super-critical and critical regimes through principal eigenvalue problems and sliding techniques, the authors establish precise decay rates and construct front-like sub- and supersolutions to trap evolving solutions. The results include a rigorous treatment of stability for front-like Cauchy data and applicability to a two-species competitive system, significantly broadening scalar-periodic insights to cooperative multi-component dynamics. This provides a solid theoretical foundation for propagation phenomena in heterogeneous periodic environments and informs front selection in ecological and physical models.

Abstract

In this paper, we consider the phenomenon of monostable pulsating fronts for multi-dimensional reaction-diffusion-advection systems in periodic media. Recent results have addressed the existence of pulsating fronts and the linear determinacy of spreading speed (Du, Li and Shen, \textit{J. Funct. Anal.} \textbf{282} (2022) 109415). In the present paper, we investigate the uniqueness and stability of monostable pulsating fronts with nonzero speed. We first derive precise asymptotic behaviors of these fronts as they approach the unstable limiting state. Utilizing these properties, we then prove the uniqueness modulo translation of pulsating fronts with nonzero speed. Furthermore, we show that these pulsating fronts are globally asymptotically stable for solutions of the Cauchy problem with front-like initial data. In particular, we establish the uniqueness and global stability of the critical pulsating front in such systems. These results are subsequently applied to a two-species competition system.

Uniqueness and stability of monostable pulsating fronts for multi-dimensional reaction-diffusion-advection systems in periodic media

TL;DR

The work extends the theory of pulsating traveling fronts to multi-component reaction-diffusion-advection systems in space-periodic media, proving uniqueness up to translation and global stability of fronts with nonzero speed. By analyzing exact front asymptotics in both super-critical and critical regimes through principal eigenvalue problems and sliding techniques, the authors establish precise decay rates and construct front-like sub- and supersolutions to trap evolving solutions. The results include a rigorous treatment of stability for front-like Cauchy data and applicability to a two-species competitive system, significantly broadening scalar-periodic insights to cooperative multi-component dynamics. This provides a solid theoretical foundation for propagation phenomena in heterogeneous periodic environments and informs front selection in ecological and physical models.

Abstract

In this paper, we consider the phenomenon of monostable pulsating fronts for multi-dimensional reaction-diffusion-advection systems in periodic media. Recent results have addressed the existence of pulsating fronts and the linear determinacy of spreading speed (Du, Li and Shen, \textit{J. Funct. Anal.} \textbf{282} (2022) 109415). In the present paper, we investigate the uniqueness and stability of monostable pulsating fronts with nonzero speed. We first derive precise asymptotic behaviors of these fronts as they approach the unstable limiting state. Utilizing these properties, we then prove the uniqueness modulo translation of pulsating fronts with nonzero speed. Furthermore, we show that these pulsating fronts are globally asymptotically stable for solutions of the Cauchy problem with front-like initial data. In particular, we establish the uniqueness and global stability of the critical pulsating front in such systems. These results are subsequently applied to a two-species competition system.
Paper Structure (12 sections, 34 theorems, 433 equations)

This paper contains 12 sections, 34 theorems, 433 equations.

Table of Contents

  1. Introduction
  2. Basic notations and assumptions
  3. Main results
  4. Preliminaries
  5. Asymptotic behavior near the unstable limiting state
  6. The super-critical case
  7. The critical case
  8. Uniqueness of pulsating traveling fronts
  9. Stability of pulsating traveling fronts
  10. The super-critical case $c>c_+^0(e)$
  11. The critical case $c=c_+^0(e)$
  12. Stability of pulsating traveling fronts

Key Result

Lemma 1.3

Assume (H1)-(H5). If $\kappa_1(\lambda_+^0)>\max_{j=2,3,\cdots,m}\kappa_j(\lambda_+^0)$, then for any $0\le\lambda\le\lambda_+^0$, problem u1um-linear-PE admits a positive periodic eigenfunction $\bm{\Phi}_{\lambda}(x)=(\phi_1^{\lambda}(x),\phi_2^{\lambda}(x),\cdots,\phi_m^{\lambda}(x))$ associated

Theorems & Definitions (71)

  • Remark 1.1
  • Definition 1.2
  • Lemma 1.3
  • proof
  • Definition 1.4
  • Remark 1.5
  • Remark 1.6
  • Theorem 1.7: see DuBao2021DuLiShen2022
  • Remark 1.8
  • Theorem 1.9: see DuLiShen2022
  • ...and 61 more