Rotating spirals in segregated reaction-diffusion systems

主 讲 人 ：Susanna Terracini    教授

活动时间：05月08日15时00分    

地      点 ：数学科学学院D204 Zoom： 672 6304 5523 密码：935785

讲座内容：

We seek rotating waves for competition-diffusion systems of the type

\partial_t u_i - \Delta u_i = \mu u_i - \beta u_i \sum_{j \neq i} a_{ij} u_j \quad \text{in } \Omega \times \mathbb{R}^+

where \Omega \subset \mathbb{R}^2 has rotational symmetry. Rigidly rotating solutions are stationary in a uniformly rotating frame. We first discuss bifurcation of rotating waves from constant ones in the case of three populations with symmetric configurations and both Dirichlet and Neumann boundary conditions.

Next, 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

\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(\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 all i and j. We also address 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 when \omega = 0.

These are joint works with Z. Li, A. Salort, G. Verzini and A. Zilio.

主讲人介绍：

Susanna Terracini, 教授，意大利都灵大学，研究方向为非线性分析及N体问题，曾获Schauder奖章，意大利Vinti Prize和Bruno Finzi Prize.

发布时间：2025-05-08 11:39:39