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Convergence of solutions of a rescaled evolution nonlocal cross-diffusion problem to its local diffusion counterpart

Gonzalo Galiano, Julián Velasco

Abstract

We prove that, under a suitable rescaling of the integrable kernel defining the nonlocal diffusion terms, the corresponding sequence of solutions of the Shigesada-Kawasaki-Teramoto nonlocal cross-diffusion problem converges to a solution of the usual problem with local diffusion. In particular, the result may be regarded as a new proof of existence of solutions for the local diffusion problem.

Convergence of solutions of a rescaled evolution nonlocal cross-diffusion problem to its local diffusion counterpart

Abstract

We prove that, under a suitable rescaling of the integrable kernel defining the nonlocal diffusion terms, the corresponding sequence of solutions of the Shigesada-Kawasaki-Teramoto nonlocal cross-diffusion problem converges to a solution of the usual problem with local diffusion. In particular, the result may be regarded as a new proof of existence of solutions for the local diffusion problem.

Paper Structure

This paper contains 4 sections, 7 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Hypothesis and main result
  3. Proof of Theorem \ref{['th:convergence']}
  4. Proofs of the lemmas

Key Result

Theorem A

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 $E_{\mathbf{u}}(t)$ defined by (def.ent), and for some constant $C>0$ independent of $J$.

Theorems & Definitions (9)

  • Theorem A: Existence and uniqueness of solution Galiano2019b
  • Theorem 1
  • Theorem B: Bourgain2001Andreu2010
  • Remark 1
  • Lemma 1
  • Corollary 1
  • Lemma 2
  • Lemma 3
  • Remark 2