[Paper Review] Pulsating solutions for multidimensional bistable and multistable equations

Thomas Giletti, Luca Rossi|arXiv (Cornell University)|Jan 22, 2019

Mathematical and Theoretical Epidemiology and Ecology Models21 references32 citations

TL;DR

The paper proves existence of pulsating travelling fronts in spatially periodic heterogeneous reaction-diffusion equations in any dimension, including both bistable and multistable settings, and shows that multistable cases may exhibit propagating terraces whose composition can depend on propagation direction.

ABSTRACT

We devote this paper to the issue of existence of pulsating travelling front solutions for spatially periodic heterogeneous reaction-diffusion equations in arbitrary dimension, in both bistable and more general multistable frameworks. In the multistable case, the notion of a single front is not sufficient to understand the dynamics of solutions, and we instead observe the appearance of a so-called propagating terrace. This roughly refers to a finite family of stacked fronts connecting intermediate stable steady states whose speeds are ordered. Surprisingly, for a given equation, the shape of this terrace (i.e., the involved intermediate states or even the cardinality of the family of fronts) may depend on the direction of propagation.

Motivation & Objective

• Motivate and formalize the existence of pulsating travelling fronts in spatially periodic media for multidimensional bistable and multistable reactions.
• Characterize the dynamics when multiple stable steady states exist, introducing the notion of propagating terraces.
• Establish conditions ensuring front existence with ordered speeds and explore directional dependence of terrace structure.

Proposed method

• Study the equation ∂t u = div(A(x)∇u) + f(x,u) with A and f periodic in space.
• Differentiate bistable and multistable regimes via Assumptions 1.1 and 1.2 and Assumption 1.3.
• Construct fronts using a time-discretization dynamical-system approach and a discrete traveling front notion.
• Show existence of monotone pulsating fronts in the bistable case (Theorem 1.4).
• Show existence of propagating terraces in the multistable case (Theorem 1.5).
• Provide an example where terrace shape depends on direction (Proposition 1.6).

Experimental results

Research questions

• RQ1 Can pulsating travelling fronts connect the extremal periodic steady states in higher-dimensional periodic media?
• RQ2 How does multistability affect front structure, and can propagating terraces arise in place of a single front?
• RQ3 Do front speeds form an ordered sequence in terraces, and how does direction influence terrace composition?
• RQ4 Under what conditions does bistable front existence extend to multidimensional periodic settings?
• RQ5 Can terrace structures differ with propagation direction in the same equation?

Key findings

• A monotone pulsating travelling front connecting the top steady state to zero exists in the bistable periodic setting (Theorem 1.4).
• In the multistable setting, a propagating terrace connecting the top state to zero exists in any direction (Theorem 1.5).
• The terrace speeds are ordered (c1 ≤ c2 ≤ ... ≤ cJ) and all intermediate states are stable (within Assumption 1.2 and 1.3).
• The shape and cardinality of the propagating terrace may depend on the propagation direction (Proposition 1.6).
• A discrete-time, iterative scheme is used to construct fronts and terraces, then passed to the continuous limit to obtain pulsating structures.