论文标题
隔离反应扩散系统中的旋转螺旋
Rotating Spirals in segregated reaction-diffusion systems
论文作者
论文摘要
我们给出了边界迹线的完整表征$φ_i$（$ i = 1，\ dots，k $）支撑螺旋波，用给定的角度速度$ω$旋转，它们看起来像是类型\ [ \ frac {\ partial} {\ partial t} u_i-Δu_i=μu_i-βu_i\ sum_i \ sum_ {j \ neq i} a_} a_ {ij} u_j} u_j} u_j \ text {in} u_i =φ_i\ text {on $ \partialΩ\ times \ times \ mathbb {r}^+$}，\ qquad u_i（\ Mathbf {X}，0）= U_ {i，0}（\ MathBf {X}）\ Text {for $ \ MathBf {X} \ inω$} \] as $β\ inω$} \]，为$β\ to +\ fo +\ infty $。这里$ω$是每$ i $和$ j $的旋转不变的平面集和$ a_ {ij}> 0 $。我们还解决了均匀的Dirichlet和Neumann边界条件，以及平面中的整个解决方案。作为我们分析的副产品，我们检测到永恒的明确家族，纯热方程式的整个解决方案，由$ω\ in \ mathbb {r} $参数化，该方程式降低至均质的谐波多项式，以$ω= 0 $。
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 $ω=0$.
