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Reaction-diffusion equations in periodic media: convergence to pulsating fronts

Hongjun Guo, François Hamel, Luca Rossi

TL;DR

This work analyzes reaction-diffusion-advection equations in spatially periodic media and proves that, under weak stability of the steady states and the existence of pulsating fronts in every direction, pulsating front profiles emerge in the large-time dynamics as local limits along sequences. It introduces and leverages the asymptotic invasion shape $ ext{W}$ and a generalized Freidlin-Gärtner formula to relate front speeds $c^*(e)$ to the geometry of $ ext{W}$, including regular and nonregular boundary points via a generalized normal set. The results cover both compact and noncompact initial data, with detailed statements on the invasion property, front convergence in directions, and the structure of $ ext{Omega}$-limit sets, along with thorough treatments of homogeneous vs periodic media. The findings generalize one-dimensional convergence to fronts to higher dimensions in periodic media, providing new insight into invasion geometry, front selection, and the large-time behavior of solutions in complex media, and offering sharp conditions under which fronts govern the asymptotics. These contributions have implications for understanding flame propagation, population dynamics, and other processes modeled by periodic RD systems in multi-dimensional settings.

Abstract

This paper is concerned with reaction-diffusion-advection equations in spatially periodic media. Under an assumption of weak stability of the constant states 0 and 1, and of existence of pulsating traveling fronts connecting them, we show that fronts' profiles appear, along sequences of times and points, in the large-time dynamics of the solutions of the Cauchy problem, whether their initial supports are bounded or unbounded. The types of equations that fit into our assumptions are the combustion and the bistable ones. We also show a generalized Freidlin-G{ä}rtner formula and other geometrical properties of the asymptotic invasion shapes, or spreading sets, of invading solutions, and we relate these sets to the upper level sets of the solutions.

Reaction-diffusion equations in periodic media: convergence to pulsating fronts

TL;DR

This work analyzes reaction-diffusion-advection equations in spatially periodic media and proves that, under weak stability of the steady states and the existence of pulsating fronts in every direction, pulsating front profiles emerge in the large-time dynamics as local limits along sequences. It introduces and leverages the asymptotic invasion shape and a generalized Freidlin-Gärtner formula to relate front speeds to the geometry of , including regular and nonregular boundary points via a generalized normal set. The results cover both compact and noncompact initial data, with detailed statements on the invasion property, front convergence in directions, and the structure of -limit sets, along with thorough treatments of homogeneous vs periodic media. The findings generalize one-dimensional convergence to fronts to higher dimensions in periodic media, providing new insight into invasion geometry, front selection, and the large-time behavior of solutions in complex media, and offering sharp conditions under which fronts govern the asymptotics. These contributions have implications for understanding flame propagation, population dynamics, and other processes modeled by periodic RD systems in multi-dimensional settings.

Abstract

This paper is concerned with reaction-diffusion-advection equations in spatially periodic media. Under an assumption of weak stability of the constant states 0 and 1, and of existence of pulsating traveling fronts connecting them, we show that fronts' profiles appear, along sequences of times and points, in the large-time dynamics of the solutions of the Cauchy problem, whether their initial supports are bounded or unbounded. The types of equations that fit into our assumptions are the combustion and the bistable ones. We also show a generalized Freidlin-G{ä}rtner formula and other geometrical properties of the asymptotic invasion shapes, or spreading sets, of invading solutions, and we relate these sets to the upper level sets of the solutions.

Paper Structure

This paper contains 32 sections, 19 theorems, 266 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. A main result with compactly supported initial conditions
  3. More general main results
  4. Some additional notations.
  5. Comments on the hypotheses and the conclusions, and complementary results
  6. Weak bistability, pulsating fronts, and invasion property
  7. Sufficient conditions in the homogeneous case.
  8. The general periodic equation \ref{['general']}, properties of pulsating traveling fronts.
  9. The invasion property.
  10. The asymptotic invasion shape: properties and examples
  11. General properties.
  12. Examples of asymptotic invasion shape for homogeneous equations with $u_0$ compactly supported.
  13. Examples for \ref{['homo']} with unbounded initial support.
  14. Counter-examples for \ref{['homo']}.
  15. Examples for the periodic equation \ref{['general']}.
  16. ...and 17 more sections

Key Result

Theorem 1.3

Assume that Hypotheses hyp:comb-bi-hyp:c*>0 hold. Let $u$ be an invading solution to general with a compactly supported initial condition $u_0$. Then, for any $e\in{\mathbb{S}}^{N-1}$, there is $\nu\in{\mathbb{S}}^{N-1}$ such that

Figures (1)

  • Figure 1: The asymptotic invasion shape $\mathcal{W}$ for $u_0=\mathbbm{1}_U$ with $U=\{x_N\leq\alpha|x'|\}$. In the case (a) $\alpha<0$, $\mathcal{W}$ satisfies the exterior and interior ball conditions at every boundary point. In the case (b) $\alpha>0$, $\mathcal{W}$ is not regular at $\bar{z}$, where $\mathcal{V}(\bar{z})=\{\nu_1,\nu_2\}$ (in dimension $N=2$).

Theorems & Definitions (36)

  • Theorem 1.3
  • Definition 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Proposition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • ...and 26 more