Diffusion, self-diffusion and cross-diffusion
type: Item
instance of: scholarly article
title: Diffusion, self-diffusion and cross-diffusion
zbMATH Open document ID: 0867.35032
author: Yuan Lou
author: Wei-Ming Ni
published in: Journal of Differential Equations
publication date: 1997-01-06T00:00:00Z
review text: The following system, which determines steady-state solutions for a corresponding parabolic system, is considered: \[ \begin{aligned} \Delta[(d_1+\alpha_{11}u_1+\alpha_{12}u_2)u_1] &+u_1(a_1-b_1u_1-c_1u_2)=0,\\ \Delta[(d_2+\alpha_{21}u_1+\alpha_{22}u_2)u_2] &+u_2(a_2-b_2u_1-c_2u_2)=0\quad\text{in }\Omega,\end{aligned} \] \[ {\partial u_1\over\partial\nu}={\partial u_2\over\partial\nu}=0\quad\text{on }\partial\Omega,\quad u_2>0,\quad u_2>0\quad\text{in }\Omega, \] where \(u_1\), \(u_2\) represent the densities of two competing species, \(\Omega\) is a bounded smooth domain of \(\mathbb{R}^N\) with \(N\geq 1\), \(\partial\Omega\) is the boundary of \(\Omega\), \(\nu\) is the outward unit normal vector on \(\partial\Omega\), \(d_i\), \(a_i\), \(b_i\), \(c_i\) \((i=1,2)\) are positive constants, \(\alpha_{ij}\) \((i,j=1,2)\) are nonnegative constants. The results of this paper are concerning the existence and the nonexistence of non-constant solutions of this system.
reviewed by: Ion Onciulescu
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35J45
zbMATH DE Number: 946925
zbMATH Keywords: existence
zbMATH Keywords: nonexistence
zbMATH Keywords: non-constant solutions
MaRDI profile type: Publication
full work available at URL: jdeq.1996.0157
OpenAlex ID: W2066496593
DOI: 10.1006/JDEQ.1996.0157
description: scientific article; zbMATH DE number 946925