Rotating Spirals in segregated reaction-diffusion systems

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
\begin{align*} \begin{cases} \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}^+\\ u_i=\varphi_i & \text{ on } \partial\Omega \times\mathbb{R}^+\\ u_i(x,0)=\varphi_i(x) & \text{ for } x \in\Omega \end{cases} \end{align*}
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 $\omega=0$.
It is a joint work with A. Salort, G. Verzini and A. Zilio.
References
[1] A. Salort, S. Terracini, G. Verzini, and A. Zilio., Rotating Spirals in segregated reaction-diffusion systems, preprint, 2022.
[2] S. Terracini, G. Verzini, and A. Zilio. Spiraling asymptotic profiles of competition-diffusion systems. Comm. Pure Appl. Math., 72(12):2578–2620, 2019.