Rotating spirals for three-component competition systems
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866913250452439040 |
|---|---|
| author | Li, Zaizheng Terracini, Susanna |
| author_facet | Li, Zaizheng Terracini, Susanna |
| contents | We investigate the existence of rotating spirals for three-component competition-diffusion systems in $B_1\subset \mathbb{R}^2$:
\begin{equation*}
\begin{cases}
\partial_tu_1-Δu_1=f(u_1)-βαu_1u_2-βγu_1 u_3,& \text{in}\ B_1\times \mathbb{R}^+,
\partial_tu_2-Δu_2=f(u_2)-βγu_1u_2-βαu_2 u_3,& \text{in}\ B_1\times \mathbb{R}^+,
\partial_tu_3-Δu_3=f(u_3)-βαu_1u_3-βγu_2 u_3,& \text{in}\ B_1\times \mathbb{R}^+,
u_i(\textbf{x},0)=u_{i,0}(\textbf{x}), i=1,2,3, &\text{in} \ B_1,
\end{cases}
\end{equation*}
with Neumann or Dirichlet boundary conditions, where $f(s)=μs(1-s)$, $μ, β>0$, $α>γ>0$. For the Neumann problem, we establish the existence of rotating spirals by applying the multi-parameter bifurcation theorem. As a byproduct, the instability of the constant positive solution is proved. In addition, for the non-homogeneous Dirichlet problem, the Rothe fixed point theorem is employed to prove the existence of rotating spirals. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_00609 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rotating spirals for three-component competition systems Li, Zaizheng Terracini, Susanna Analysis of PDEs 35K51, 35B32, 92D25 We investigate the existence of rotating spirals for three-component competition-diffusion systems in $B_1\subset \mathbb{R}^2$: \begin{equation*} \begin{cases} \partial_tu_1-Δu_1=f(u_1)-βαu_1u_2-βγu_1 u_3,& \text{in}\ B_1\times \mathbb{R}^+, \partial_tu_2-Δu_2=f(u_2)-βγu_1u_2-βαu_2 u_3,& \text{in}\ B_1\times \mathbb{R}^+, \partial_tu_3-Δu_3=f(u_3)-βαu_1u_3-βγu_2 u_3,& \text{in}\ B_1\times \mathbb{R}^+, u_i(\textbf{x},0)=u_{i,0}(\textbf{x}), i=1,2,3, &\text{in} \ B_1, \end{cases} \end{equation*} with Neumann or Dirichlet boundary conditions, where $f(s)=μs(1-s)$, $μ, β>0$, $α>γ>0$. For the Neumann problem, we establish the existence of rotating spirals by applying the multi-parameter bifurcation theorem. As a byproduct, the instability of the constant positive solution is proved. In addition, for the non-homogeneous Dirichlet problem, the Rothe fixed point theorem is employed to prove the existence of rotating spirals. |
| title | Rotating spirals for three-component competition systems |
| topic | Analysis of PDEs 35K51, 35B32, 92D25 |
| url | https://arxiv.org/abs/2403.00609 |