# Existence and instability of steady states for a triangular cross-diffusion system: a computer-assisted proof

## TL;DR

This paper introduces a computer-assisted method to rigorously prove the existence and analyze the stability of steady states in a triangular cross-diffusion system, revealing multiple coexisting states and their instability.

## Contribution

It develops an a posteriori validation technique based on fixed point arguments for studying steady states and their stability in complex cross-diffusion systems.

## Key findings

- Existence of multiple non homogeneous steady states, up to 13 coexisting.
- Many steady states are proven to be unstable.
- Method successfully combines numerical computation with rigorous proof.

## Abstract

In this paper, we present and apply a computer-assisted method to study steady states of a triangular cross-diffusion system. Our approach consist in an a posteriori validation procedure, that is based on using a fxed point argument around a numerically computed solution, in the spirit of the Newton-Kantorovich theorem. It allows us to prove the existence of various non homogeneous steady states for different parameter values. In some situations, we get as many as 13 coexisting steady states. We also apply the a posteriori validation procedure to study the linear stability of the obtained steady states, proving that many of them are in fact unstable.