# On a Class of Singularly Perturbed Elliptic Systems with Asymptotic Phase Segregation

## TL;DR

This paper investigates singularly perturbed elliptic systems modeling phase segregation, proving existence, uniqueness, and asymptotic convergence to a free boundary problem with novel explicit solutions and numerical simulations.

## Contribution

It introduces a new method for solving the limiting free boundary problem explicitly for specific parameters, advancing understanding of phase segregation models.

## Key findings

- Proved existence and uniqueness of solutions.
- Established convergence to a free boundary limit.
- Provided explicit solutions and numerical simulations.

## Abstract

This work is devoted to study of a class of elliptic singular perturbed systems and their singular limit to a phase segregating system. We prove existence and uniqueness and study the asymptotic behaviour with convergence to a limiting problem as the interaction rate tends to infinity. The limiting problem is a free boundary problem such that at each point in the domain at least one of the components is zero which implies simultaneously all components can not coexist. We present a novel method, which provides an explicit solution of limiting problem for special choice of parameters. Moreover, we present some numerical simulations of the asymptotic problem.