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Stability of propagating terraces in spatially periodic multistable equations in $\mathbb{R}^N$

Thomas Giletti, Luca Rossi

TL;DR

This work analyzes large-time dynamics of multistable reaction-diffusion equations in spatially periodic media, revealing that solutions converge to a unique propagating terrace in each direction and that spreading is governed by terrace speeds through Wulff shapes. The authors develop a planar-case framework to prove terrace uniqueness, construct an algorithm to identify terrace platforms, and connect these terraces to planar-like and compactly supported initial data. Under the same_shape and regular Wulff-shape assumptions, they obtain sharp Freidlin–Gärtner-type descriptions of spreading, while they also discuss limitations and possible non-smoothness in Wulff shapes arising from directional heterogeneity. The results extend classical bistable and homogeneous theories to higher dimensions and heterogeneous media, providing a detailed geometric view of invasion via propagating terraces.

Abstract

In this paper, we study the large time behaviour of solutions of multistable reaction-diffusion equations in $\mathbb{R}^N$, with a spatially periodic heterogeneity. By multistable, we mean that the problem admits a finite -- but arbitrarily large -- number of stable, periodic steady states. In contrast with the more classical monostable and bistable frameworks, which exhibit the emergence of a single travelling front in the long run, in the present case the large time dynamics is governed by a family of stacked travelling fronts, involving intermediate steady states, called propagating terrace. Their existence in the multidimensional case has been established in our previous work [13]. The first result of the present paper is their uniqueness. Next, we show that the speeds of the propagating terraces in different directions dictate the spreading speeds of solutions of the Cauchy problem, for both planar-like and compactly supported initial data. The latter case turns out to be much more intricate than the former, due to the fact that the propagating terraces in distinct directions may involve different sets of intermediate steady states. Another source of difficulty is that the Wulff shape of the speeds of travelling fronts can be non-smooth, as we show in the bistable case using a result of [4].

Stability of propagating terraces in spatially periodic multistable equations in $\mathbb{R}^N$

TL;DR

This work analyzes large-time dynamics of multistable reaction-diffusion equations in spatially periodic media, revealing that solutions converge to a unique propagating terrace in each direction and that spreading is governed by terrace speeds through Wulff shapes. The authors develop a planar-case framework to prove terrace uniqueness, construct an algorithm to identify terrace platforms, and connect these terraces to planar-like and compactly supported initial data. Under the same_shape and regular Wulff-shape assumptions, they obtain sharp Freidlin–Gärtner-type descriptions of spreading, while they also discuss limitations and possible non-smoothness in Wulff shapes arising from directional heterogeneity. The results extend classical bistable and homogeneous theories to higher dimensions and heterogeneous media, providing a detailed geometric view of invasion via propagating terraces.

Abstract

In this paper, we study the large time behaviour of solutions of multistable reaction-diffusion equations in , with a spatially periodic heterogeneity. By multistable, we mean that the problem admits a finite -- but arbitrarily large -- number of stable, periodic steady states. In contrast with the more classical monostable and bistable frameworks, which exhibit the emergence of a single travelling front in the long run, in the present case the large time dynamics is governed by a family of stacked travelling fronts, involving intermediate steady states, called propagating terrace. Their existence in the multidimensional case has been established in our previous work [13]. The first result of the present paper is their uniqueness. Next, we show that the speeds of the propagating terraces in different directions dictate the spreading speeds of solutions of the Cauchy problem, for both planar-like and compactly supported initial data. The latter case turns out to be much more intricate than the former, due to the fact that the propagating terraces in distinct directions may involve different sets of intermediate steady states. Another source of difficulty is that the Wulff shape of the speeds of travelling fronts can be non-smooth, as we show in the bistable case using a result of [4].

Paper Structure

This paper contains 19 sections, 26 theorems, 256 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Statement of the main results
  3. Uniqueness of the propagating terrace
  4. Planar-like initial data
  5. Compactly supported initial data
  6. Plan of the paper.
  7. The planar case
  8. Preliminaries
  9. Uniqueness of the propagating terrace
  10. A procedure for the determination of the terrace
  11. Spreading speeds for planar-like initial data
  12. The compactly supported case under Assumption \ref{['ass:same_shape']}
  13. An upper bound for the spreading speeds
  14. A lower bound for the speed towards $\overline{p}$
  15. Conclusion under Assumption \ref{['ass:same_shape']}
  16. ...and 4 more sections

Key Result

Theorem 1.3

Under Assumption ass:multi, for any $e \in S^{N-1}$, there exists a propagating terrace $\mathcal{T} = ((q_k),(U_k))$ connecting $\overline{p}$ to $0$ in the direction $e$. Furthermore, each platform $q_k$ of this terrace is a stable, periodic, steady state, and each profile $U_k(x,z)$ is nonincreas

Figures (1)

  • Figure 1: Illustration of the construction of the supersolution $\overline{u}^\eta$ at a fixed time $t$. The horizontal axis represents the variable $x \cdot e$. The horizontal straight lines are the steady states (including three platforms). The solid line is $\overline{u}^\eta$, which is taken in each interval as a minimum between two perturbed travelling fronts (drawn in either dashed or dash-dotted lines).

Theorems & Definitions (57)

  • Definition 1.1: Pulsating travelling front
  • Definition 1.2: Propagating terrace
  • Theorem 1.3: GR_2020
  • Remark 1.1
  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.1
  • Definition 2.3: Wulff shape
  • Theorem 2.4
  • Definition 2.5
  • ...and 47 more