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Front propagation into unstable states for periodic monotone reaction-diffusion systems

Liangliang Deng, Arnaud Ducrot, Quentin Griette

TL;DR

This work analyzes front propagation into unstable states for multi-component, spatially periodic cooperative reaction–diffusion systems in arbitrary dimensions. By combining principal eigenvalue analysis of the linearized operator with a direction-dependent coordinate framework and a constructive fixed-point approach, it establishes speeds obeying $c \ge c^*(e)$, linear determinacy under sublinear nonlinearities, and monotonicity of waves under subhomogeneity; it also provides a rigorous existence theory for pulsating waves in all directions via rational-direction reduction and cylinder-based fixed-point arguments, plus a nonexistence regime when the trivial state is linearly stable and the nonlinearity is strictly sublinear. The results extend front-propagation theory to multi-component monotone systems in periodic media and offer a robust, direction-aware construction mechanism with hair-trigger persistence underpinning the asymptotic behavior. Overall, the paper clarifies how linearization around the unstable state and nonlinear structure govern the existence and speed of pulsating fronts in higher dimensions.

Abstract

In this paper we study the invasion fronts of spatially periodic monotone reaction-diffusion systems in a multi-dimensional setting. We study the pulsating traveling waves that connect the trivial equilibrium, for which all components of the state variable are identically equal to zero, to a uniformly persistent stationary state, for which all components are uniformly positive. When the trivial equilibrium is linearly unstable, we show that all pulsating traveling waves have a speed that is greater than the speed of the linearized system at the equilibrium, in any given direction. If moreover the nonlinearity is sublinear, then we can construct a pulsating traveling wave that travels at any super-linear speed in any given direction (i.e. the minimal speed is linearly determined). We also show that pulsating traveling waves are monotonic in time as soon as the nonlinearity is sub-homogeneous. Beyond these general qualitative properties, the main focus of the paper is to derive sufficient conditions for the existence and nonexistence of pulsating waves propagating in any given direction. Our proof of the existence part relies upon a new level of understanding of the multi-dimensional pulsating waves observed from a direction-dependent coordinate system.

Front propagation into unstable states for periodic monotone reaction-diffusion systems

TL;DR

This work analyzes front propagation into unstable states for multi-component, spatially periodic cooperative reaction–diffusion systems in arbitrary dimensions. By combining principal eigenvalue analysis of the linearized operator with a direction-dependent coordinate framework and a constructive fixed-point approach, it establishes speeds obeying , linear determinacy under sublinear nonlinearities, and monotonicity of waves under subhomogeneity; it also provides a rigorous existence theory for pulsating waves in all directions via rational-direction reduction and cylinder-based fixed-point arguments, plus a nonexistence regime when the trivial state is linearly stable and the nonlinearity is strictly sublinear. The results extend front-propagation theory to multi-component monotone systems in periodic media and offer a robust, direction-aware construction mechanism with hair-trigger persistence underpinning the asymptotic behavior. Overall, the paper clarifies how linearization around the unstable state and nonlinear structure govern the existence and speed of pulsating fronts in higher dimensions.

Abstract

In this paper we study the invasion fronts of spatially periodic monotone reaction-diffusion systems in a multi-dimensional setting. We study the pulsating traveling waves that connect the trivial equilibrium, for which all components of the state variable are identically equal to zero, to a uniformly persistent stationary state, for which all components are uniformly positive. When the trivial equilibrium is linearly unstable, we show that all pulsating traveling waves have a speed that is greater than the speed of the linearized system at the equilibrium, in any given direction. If moreover the nonlinearity is sublinear, then we can construct a pulsating traveling wave that travels at any super-linear speed in any given direction (i.e. the minimal speed is linearly determined). We also show that pulsating traveling waves are monotonic in time as soon as the nonlinearity is sub-homogeneous. Beyond these general qualitative properties, the main focus of the paper is to derive sufficient conditions for the existence and nonexistence of pulsating waves propagating in any given direction. Our proof of the existence part relies upon a new level of understanding of the multi-dimensional pulsating waves observed from a direction-dependent coordinate system.

Paper Structure

This paper contains 17 sections, 17 theorems, 275 equations.

Table of Contents

  1. Introduction and main results
  2. Setting of the problem
  3. Statement of the results
  4. Elements of the proof
  5. Examples
  6. Preliminaries
  7. Qualitative properties of pulsating waves
  8. Lower bound on the wave speed
  9. Monotonicity of the wave with respect to time
  10. Existence of pulsating waves
  11. Construction of two sub- and supersolutions
  12. Description of a direction-dependent coordinate system
  13. Construction of the wave in each rational direction
  14. Completion of the proof of Theorem \ref{['main-result-existence']}
  15. Nonexistence of pulsating waves
  16. ...and 2 more sections

Key Result

Theorem 1.1

Assume $\lambda^p_{1}<0$. Let $(c,u)$ be a pulsating travelling wave solution to monotone-system. Then the wave speed $c$ satisfies If $f$ is further assumed to be subhomogeneous, i.e. then the function $u=u(t,x)$ is componentwise increasing in $t$ if $c>0$ and decreasing if $c<0$. Moreover, one has $c\partial_t u(t,x)\gg\bm{0}$ for all $(t,x)\in\mathbb{R}\times\mathbb{R}^N$.

Theorems & Definitions (38)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Proposition 2.1
  • Proposition 2.2
  • ...and 28 more