Rotating Spirals in segregated reaction-diffusion systems
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866909524685750272 |
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| author | Salort, Ariel Terracini, Susanna Verzini, Gianmaria Zilio, Alessandro |
| author_facet | Salort, Ariel Terracini, Susanna Verzini, Gianmaria Zilio, Alessandro |
| contents | 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 \[
\frac{\partial}{\partial t} u_i -Δu_i = μu_i -βu_i \sum_{j \neq i} a_{ij} u_j \text{ in } Ω\times\mathbb{R}^+, \qquad
u_i = φ_i \text{ on $\partialΩ\times\mathbb{R}^+$}, \qquad
u_i(\mathbf{x},0) = u_{i,0}(\mathbf{x}) \text{ for $\mathbf{x} \in Ω$} \] as $β\to +\infty$. Here $Ω$ 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 $ω\in\mathbb{R}$, which reduce to homogeneous harmonic polynomials for $ω=0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_10369 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Rotating Spirals in segregated reaction-diffusion systems Salort, Ariel Terracini, Susanna Verzini, Gianmaria Zilio, Alessandro Analysis of PDEs 35B25, 35B36, 35K51, 92D25 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 \[ \frac{\partial}{\partial t} u_i -Δu_i = μu_i -βu_i \sum_{j \neq i} a_{ij} u_j \text{ in } Ω\times\mathbb{R}^+, \qquad u_i = φ_i \text{ on $\partialΩ\times\mathbb{R}^+$}, \qquad u_i(\mathbf{x},0) = u_{i,0}(\mathbf{x}) \text{ for $\mathbf{x} \in Ω$} \] as $β\to +\infty$. Here $Ω$ 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 $ω\in\mathbb{R}$, which reduce to homogeneous harmonic polynomials for $ω=0$. |
| title | Rotating Spirals in segregated reaction-diffusion systems |
| topic | Analysis of PDEs 35B25, 35B36, 35K51, 92D25 |
| url | https://arxiv.org/abs/2202.10369 |