Uniqueness result for long range spatially segregation elliptic system

TL;DR

This paper investigates a class of elliptic competition-diffusion systems modeling long-range species segregation, establishing existence, uniqueness, and the limiting behavior as competition intensifies, leading to free boundary problems.

Contribution

It proves the existence and uniqueness of solutions for long-range segregation models and characterizes their limiting behavior as competition increases.

Findings

Solutions exist and are unique for the models studied.

As competition rate increases, solutions converge to a segregated state.

The limiting segregated state satisfies specific free boundary conditions.

Abstract

We study a class of elliptic competition-diffusion systems of long range segregation models for two and more competing species. The existence and uniqueness of the solution are shown. We prove that as the competition rate goes to infinity the solution converges, along with suitable sequences, to a spatially long range segregated state satisfying some free boundary problems.