Spirali rotanti in sistemi di reazione-diffusione segregati.

In this dissertation we aim to describe, under suitable assumptions, the nodal set of $K \geq 3$ competing species in $N=2$ dimensions. In particular, we prove that spiraling profiles arise as the structure of the nodal set in the singular limit, i.e. when the interspecific competition rates become infinite, of segregated reaction-diffusion systems with strongly competitive interaction of Lotka-Volterra type. For this purpose, let $u_{i}$ be the (non negative) densities of the above species and let \begin{equation*} \begin{cases}-\Delta u_{i}(x)=- \beta u_{i}(x) \sum_{j \neq i} a_{i j} u_{j}(x) & x \in \Omega, \\ u_{i}(x)=\varphi_{i}(x) & x \in \partial \Omega\end{cases} \end{equation*} be the stationary reaction-diffusion system considered, where we can assume $\Omega$ is the unit ball. The coefficients $\beta a_{ij}$ account for the interspecific competition rates; they are positive so that the interaction has a repulsive character. We prove that, denoting by $u_{i,\beta}$ a solution of the above system at fixed $\beta$, the sequence $u_{i,\beta}$ as $\beta$ tends to infinity converges to a limit profile $u_{i}$ for every $i$ and the limit densities segregate, i.e. $u_{i}\cdot u_{j} \equiv 0$ for every $i \neq j$. The surfaces between two different positivity sets of two densities represents the free boundary we want to describe. Our goal is to describe the behavior of these surfaces near a singular point with multiplicity $h \geq 3$. Our study make use of the conformal transformation $$ \begin{aligned} \mathcal{T}:(x, y) \mapsto x=\left(e^{-y} \cos x, e^{-y} \sin x\right), \end{aligned} $$ from the half-plane to the punctured disk in order to translate the problem into $\mathbb{R} \times \mathbb{R}^{+}$. Let us call $v$ a suitable combination of the transformed densities. Thanks to the above map, we are able to find a description of the nodal lines of $v$, which turn out to be asymptotic to straight lines. Using again $\mathcal{T}$, we map back $v$ and its nodal set and we obtain that the the transformed straight lines are actually logarithmic curves spiraling toward the singular point. Then we tackle the time-dependent problem. To this aim we consider the following system \begin{equation*} \begin{cases}\partial_{t} u_{i}-\Delta u_{i}=f_{i}\left(u_{i}\right)-\beta u_{i} \sum_{j \neq i} a_{i j} 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 \end{cases} \end{equation*} We want to prove that the behavior of nodal curves of the limit profiles near a singular point can be described by logarithmic rotating spirals. In order to do so, we assume that $\Omega$ is rotating with frequency $\omega$, so that we can reduce ourselves to look for rigidly rotating logarithmic spirals which are steady states of a spinning frame. Again, the conformal map plays a fundamental role: we first solve a suitable problem in the half-plane for a specific function $v$ and find the behavior of its nodal set, then we prove that, under suitable conditions, $v$ has the appropriate nodal properties to be mapped back to the disk. We find that the nodal lines of $v$ are again asymptotic to straight lines; therefore, applying $\mathcal{T}$, we obtain that the behavior of the nodal curves of the limit densities can be described by logarithmic spirals near a singular point.