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Research Article | Open Access

Martingale solutions of stochastic nonlocal cross-diffusion systems

Institut de Mathématiques de Bordeaux UMR CNRS 525, Université de Bordeaux, F-33076 Bordeaux Cedex, France
Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern, N–0316 Oslo, Norway

The work of Bendahmane is supported by the Fincome project

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Abstract

We establish the existence of solutions for a class of stochastic reaction-diffusion systems with cross-diffusion terms modeling interspecific competition between two populations. More precisely, we prove the existence of weak martingale solutions employing appropriate Faedo-Galerkin approximations and the stochastic compactness method. The nonnegativity of solutions is proved by a stochastic adaptation of the well-known Stampacchia approach.

Keywords

Stochastic partial differential equationcross-diffusion systemexistencemartingale solutionpopulation dynamics
CLC number: Primary: 60H15, 35K57; Secondary: 35M10, 35A05

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Networks and Heterogeneous Media
Volume 17 Issue 5,
October 2022
Pages 719-752
DOI: 10.3934/nhm.2022024

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Cite this article:
Bendahmane M, Karlsen KH. Martingale solutions of stochastic nonlocal cross-diffusion systems. Networks and Heterogeneous Media, 2022, 17(5): 719-752. https://doi.org/10.3934/nhm.2022024

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Received: 01 March 2022
Published: 15 October 2022
©2022 the Author(s), licensee AIMS Press.

This is an open access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0)