Rotating Spirals in segregated reaction-diffusion systems

TL;DR

This paper characterizes boundary conditions supporting spiraling wave solutions in reaction-diffusion systems, analyzing their limits as competition intensifies, and identifies explicit eternal solutions of the heat equation related to harmonic polynomials.

Contribution

It provides a complete characterization of boundary traces supporting spiraling waves in reaction-diffusion systems with various boundary conditions and limits.

Findings

Explicit families of eternal solutions to the heat equation are identified.

Spiraling waves are characterized as limits of competition-diffusion systems.

Connections to harmonic polynomials are established for zero angular speed.

Abstract

We give a complete characterization of the boundary traces $\varphi_i$ ($i=1,\dots,K$) supporting spiraling waves, rotating with a given angular speed $\omega$, which appear as singular limits of competition-diffusion systems of the type \[ \frac{\partial}{\partial t} u_i -\Delta u_i = \mu u_i -\beta u_i \sum_{j \neq i} a_{ij} u_j \text{ in } \Omega \times\mathbb{R}^+, \qquad u_i = \varphi_i \text{ on $\partial\Omega\times\mathbb{R}^+$}, \qquad u_i(\mathbf{x},0) = u_{i,0}(\mathbf{x}) \text{ for $\mathbf{x} \in \Omega$} \] as $\beta\to +\infty$. Here $\Omega$ is a rotationally invariant planar set and $a_{ij}>0$ for every $i$ and $j$. We tackle also the homogeneous Dirichlet and Neumann boundary conditions, as well as entire solutions in the plane. As a byproduct of our analysis we detect explicit families of eternal, entire solutions of the pure heat equation, parameterized by $\omega\in\mathbb{R}$, which reduce to homogeneous harmonic polynomials for $\omega=0$.