A variational problem for the spatial segregation of reaction--diffusion systems

TL;DR

This paper investigates stationary states of reaction-diffusion systems with multiple densities, establishing existence, uniqueness conditions, and analyzing the regularity and properties of their segregated supports.

Contribution

It introduces a variational framework for analyzing spatial segregation in reaction-diffusion systems with three or more densities, including existence, uniqueness, and regularity results.

Findings

Proved existence of segregation states under a variational principle.

Derived conditions for the uniqueness of these states.

Analyzed regularity and qualitative properties of densities and free boundaries.

Abstract

In this paper we study a class of stationary states for reaction--diffusion systems of $k\geq 3$ densities having disjoint supports. For a class of segregation states governed by a variational principle we prove existence and provide conditions for uniqueness. Some qualitative properties and the local regularity both of the densities and of their free boundaries are established in the more general context of a functional class characterized by differential inequalities.