On some model problem for the propagation of interacting species in a special environment

Michel Chipot; Mingmin Zhang

doi:10.3934/dcds.2020401

Article Contents

2021, Volume 41, Issue 7: 3141-3161. Doi: 10.3934/dcds.2020401

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On some model problem for the propagation of interacting species in a special environment

Michel Chipot^1, and
Mingmin Zhang^2,3, ,

1.
Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, 8057 Zürich, Switzerland
2.
School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, China
3.
Aix Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, 13453 Marseille, France

^* Corresponding author: Mingmin Zhang
^* Corresponding author: Mingmin Zhang

Received: March 31, 2020

Revised: September 30, 2020

Early access: December 2020

Published: July 2021

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Abstract

The purpose of this note is to study the existence of a nontrivial solution for an elliptic system which comes from a newly introduced mathematical problem so called Field-Road model. Specifically, it consists of coupled equations set in domains of different dimensions together with some interaction of non classical type. We consider a truncated problem by imposing Dirichlet boundary conditions and an unbounded setting as well.

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Mathematics Subject Classification: 35J47, 35J57.

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Figure 1. The domain $ \Omega_\ell $ for one-road problem

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Figure 2. The domain $ \Omega_\ell $ for two-road problem

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Figure 3. The graph of the function $ \rho(x_1) $

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