Rotating spirals in segregated reaction-diffusion systems

Salort, Ariel; Terracini, Susanna; Verzini, Gianmaria; Zilio, Alessandro

doi:10.2140/apde.2025.18.549

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Rotating spirals in segregated reaction-diffusion systems

Ariel Salort, Susanna Terracini, Gianmaria Verzini and Alessandro Zilio

Vol. 18 (2025), No. 3, 549–590

DOI: 10.2140/apde.2025.18.549

Abstract

We give a complete characterization of the boundary traces $φ_{i}$ ( $i = 1, \dots, K$ ) supporting spiraling waves, rotating with a given angular speed $ω$ , which appear as singular limits of competition-diffusion systems of the type

{\begin{matrix} \partial_{t} u_{i} - Δ u_{i} = μ u_{i} - β u_{i} \sum_{j \neq i} a_{i j} u_{j} & in Ω \times ℝ^{+}, \\ u_{i} = φ_{i} & on \partial Ω \times ℝ^{+}, \\ u_{i} (x, 0) = u_{i, 0} (x) & for x \in Ω, \end{matrix}

as $β \to + \infty$ . Here $Ω$ is a rotationally invariant planar set, and $a_{i j} > 0$ 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, parametrized by $ω \in ℝ$ , which reduce to homogeneous harmonic polynomials for $ω = 0$ .

Keywords

competition-diffusion systems, singular perturbation, free boundary problems, spiral waves

Mathematical Subject Classification

Primary: 35B25, 35B36

Secondary: 35K51, 92D25

Milestones

Received: 21 February 2022

Revised: 1 September 2023

Accepted: 21 November 2023

Published: 3 March 2025

Authors

Ariel Salort
Instituto de Cálculo, FCEyN Universidad de Buenos Aires Ciudad Universitaria Buenos Aires Argentina

Susanna Terracini
Dipartimento di Matematica “Giuseppe Peano” Università di Torino Torino Italy

Gianmaria Verzini
Dipartimento di Matematica Politecnico di Milano Milano Italy

Alessandro Zilio
Laboratoire Jacques Louis Lions (CNRS UMR 7598) Université Paris Diderot Paris France

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Vol. 18, No. 3, 2025