On a long range segregation model

TL;DR

This paper investigates a non-local segregation model involving a family of equations with a small parameter, analyzing how populations segregate at a certain distance under various non-local interactions.

Contribution

It introduces a new class of non-local segregation equations with explicit models and analyzes their properties as the parameter tends to zero.

Findings

Populations tend to segregate at a fixed distance in the limit.

The model captures non-local interactions such as averaging and supremum.

Mathematical analysis of the limiting behavior as epsilon approaches zero.

Abstract

In this work we study the properties of segregation processes modeled by a family of equations $$ L(u_i) (x) = u_i(x)\: F_i (u_1, \ldots, u_K)(x)\qquad i=1,\ldots, K $$ where $F_i (u_1, \ldots, u_K)(x)$ 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 $$\Delta u_i^\ep (x) = \frac1{\ep^2} u_i^\ep (x)\sum_{i\neq j} H(u_j^\ep)(x)$$ 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 $\real^n$. Heuristically, this will force the populations to stay at $\rho$-distance 1, one from each other, as $\ep\to0$.