# Well-posedness of a cross-diffusion population model with nonlocal diffusion

## TL;DR

This paper establishes the mathematical well-posedness, including existence and uniqueness, of a nonlocal cross-diffusion population model that generalizes classical models by replacing differential operators with integral operators.

## Contribution

It introduces a nonlocal version of the cross-diffusion model and proves fundamental mathematical properties, expanding the theoretical understanding of such population dynamics models.

## Key findings

- Existence of solutions proven using compactness arguments.
- Uniqueness of solutions established via duality techniques.
- Model serves as an approximation to classical cross-diffusion models.

## Abstract

We prove the existence and uniqueness of solution of a nonlocal cross-diffusion competitive population model for two species. The model may be considered as a version, or even an approximation, of the paradigmatic Shigesada-Kawasaki-Teramoto cross-diffusion model, in which the usual diffusion differential operator is replaced by an integral diffusion operator. The proof of existence of solutions is based on a compactness argument, while the uniqueness of solution is achieved through a duality technique.