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

Gonzalo Galiano, Julián Velasco

TL;DR

The existence and uniqueness of solution of a nonlocal cross-diffusion competitive population model for two species is proved using a compactness argument and a duality technique.

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.

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

TL;DR

The existence and uniqueness of solution of a nonlocal cross-diffusion competitive population model for two species is proved using a compactness argument and a duality technique.

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.

Paper Structure

This paper contains 7 sections, 3 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Assumptions and main results
  3. Existence of solutions of a regularized and approximated problem
  4. Existence of solutions of a time independent problem
  5. Passing to the limit $\tau\to 0$
  6. Passing to the limit $\varepsilon\to0$
  7. Uniqueness of solution

Key Result

Theorem 1

Assume (H) and Then, there exists a unique strong solution $(u_1,u_2)$ of problem (eq.eq)-(eq.id) with $u_i\geq 0$ a.e. in $Q_T$ and such that, for $i=1,2$ and $t\in [0,T]$, with $E(t)$ defined by (def.ent), and for some constant $c>0$ independent of $J$.

Theorems & Definitions (10)

  • Remark 1
  • Theorem 1
  • Remark 2
  • Remark 3
  • Lemma 1
  • proof
  • Remark 4
  • Corollary 1
  • proof
  • Remark 5