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Curved fronts of bistable reaction-diffusion equations in spatially periodic media: $N\ge 2$

Hongjun Guo, Haijian Wang

TL;DR

This work analyzes curved transition fronts for the spatially periodic bistable reaction-diffusion equation in $N\ge 2$ dimensions. Under the a priori assumption that pulsating fronts exist in every direction, the authors glue finitely many such fronts along polyhedral asymptotic planes to produce polytope-like curved fronts with interfaces $\Gamma_t=\partial\mathcal{Q}+\hat{c}t e_0$, proving existence, uniqueness, and asymptotic stability; they also construct fronts with reversed zone assignments. The approach relies on a sub-/supersolution framework built from pulsating fronts and a surface with asymptotic planes, together with a generalized comparison principle. This advances the theory of transition fronts in spatially periodic media by enriching front geometry beyond planar or conical shapes to polyhedral-like structures, while ensuring robustness to perturbations. The results have implications for understanding front propagation in heterogeneous periodic environments and broaden the toolkit for analyzing curved fronts in higher dimensions.

Abstract

This paper is concerned with curved fronts of bistable reaction-diffusion equations in spatially periodic media for dimensions $N\geq 2$. The curved fronts concerned are transition fronts connecting $0$ and $1$. Under a priori assumption that there exist moving pulsating fronts in every direction, we show the existence of polytope-like curved fronts with $0$-zone being a polytope and $1$-zone being the complementary set. By reversing some conditions, we also show the existence of curved fronts with reversed $0$-zone and $1$-zone. Furthermore, the curved fronts constructed by us are proved to be unique and asymptotic stable.

Curved fronts of bistable reaction-diffusion equations in spatially periodic media: $N\ge 2$

TL;DR

This work analyzes curved transition fronts for the spatially periodic bistable reaction-diffusion equation in dimensions. Under the a priori assumption that pulsating fronts exist in every direction, the authors glue finitely many such fronts along polyhedral asymptotic planes to produce polytope-like curved fronts with interfaces , proving existence, uniqueness, and asymptotic stability; they also construct fronts with reversed zone assignments. The approach relies on a sub-/supersolution framework built from pulsating fronts and a surface with asymptotic planes, together with a generalized comparison principle. This advances the theory of transition fronts in spatially periodic media by enriching front geometry beyond planar or conical shapes to polyhedral-like structures, while ensuring robustness to perturbations. The results have implications for understanding front propagation in heterogeneous periodic environments and broaden the toolkit for analyzing curved fronts in higher dimensions.

Abstract

This paper is concerned with curved fronts of bistable reaction-diffusion equations in spatially periodic media for dimensions . The curved fronts concerned are transition fronts connecting and . Under a priori assumption that there exist moving pulsating fronts in every direction, we show the existence of polytope-like curved fronts with -zone being a polytope and -zone being the complementary set. By reversing some conditions, we also show the existence of curved fronts with reversed -zone and -zone. Furthermore, the curved fronts constructed by us are proved to be unique and asymptotic stable.
Paper Structure (8 sections, 21 theorems, 225 equations, 2 figures)

This paper contains 8 sections, 21 theorems, 225 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Properties of pulsating fronts and a comparison principle
  4. A surface with asymptotic planes
  5. Supersolutions and subsolutions
  6. Existence of curved fronts
  7. Stability of curved fronts
  8. Appendix

Key Result

Theorem 1.3

For any $\{e_i\}_{i=1}^n$ of $\mathbb{S}^{N-1}$ such that there exists a transition front $V(t,x)$ of 1.1 with $\Gamma_t$, $\Omega_t^{\pm}$ given by satisfying $V_t(t,x)>0$ for all $(t,x)\in\mathbb{R}\times\mathbb{R}^N$ and Moreover, such a transition front is unique in the sense that if an entire solution $\widetilde{V}(t,x)$ satisfies thm-existence-1, then $\widetilde{V}(t,x)\equiv V(t,x)$.

Figures (2)

  • Figure 1: 2-dimensional curved fronts in Theorem \ref{['thm-existence']} and Theorem \ref{['coro']}.
  • Figure 2: an example of $\mathcal{Q}_i$ and $y=\varphi_i(x)$ in dimension 2.

Theorems & Definitions (24)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Definition 2.4
  • ...and 14 more