On a long range segregation model

Abstract

In this work we study the properties of segregation processes modeled by a family of equations $$L(u_i)(x)=u_i(x)F_i(u_1,\dots ,u_K)(x)i=1,\dots ,K$$ where $F_i(u_1,\dots ,u_K)(x)$ is a non-local factor that takes into consideration the values of the functions $u_j$'s in a full neighborhood of $x.$ We consider as a model problem $$Δu_i^\ep (x) = \frac1{\ep^2} u_i^\ep (x)\sum_{i\neq j} H(u_j^\ep)(x)$$ where $\ep$ is a small parameter and $H(u_j^\ep)(x)$ is for instance $$H(u_j^\ep)(x)= \int_{\mathcal{B}_1 (x)} u_j^\ep (y)\, \text{d}y$$ or $$H(u_j^\ep)(x)= \sup_{y\in \mathcal{B}_1(x)} u_j^\ep (y).$$ Here the set ${\mathcal{B}}_1(x)$ is the unit ball centered at $x$ with respect to a smooth, uniformly convex norm $\rho$ of ${\Re }^n$. Heuristically, this will force the populations to stay at $\rho$-distance 1, one from each other, as $\ep\to0$.