On stable space dependent stationary solutions of a competition system with diffusion
type: Item
instance of: scholarly article
title: On stable space dependent stationary solutions of a competition system with diffusion
zbMATH Open document ID: 0533.35052
DOI: 10.4171/ZAA/59
author: Andrea Schiaffino
author: Piero De Mottoni
author: Alberto Tesei
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 1983-01-01T00:00:00Z
review text: A quasilinear parabolic system of two equations is considered in connection with the following initial boundary value problem: \[ \partial_t u = a_1\Delta u + u(b_1 - c_1u - d_1v),\quad \partial_t v = v(b_2 - c_2v - d_2u) \quad\text{in } (0,+\infty)\times \Omega, \] \[ u=0\quad\text{in } (0,+\infty)\times \partial \Omega, \quad u=u_0,\ v=v_0\quad\text{in }\{0\} \times \Omega. \] Here \(\Omega \subset \mathbb R^n\) is an open bounded set with smooth boundary \(\partial \Omega\), \(a_i,b_i,c_i,d_i\) \((i=1,2)\) are positive constants and \(u_0\), \(v_0\) are given nonnegative functions. The authors investigate existence, uniqueness and asymptotic stability of the following elliptic problem: \[ a_1\Delta u + u(b_1 - c_1u - d_1v) = 0,\quad v(b_2 - c_2v - d_2u) = 0\quad\text{in } \Omega, \ u=0\quad\text{in } \partial \Omega. \] The main purpose of the paper is to prove what happens when arbitrarily small diffusion is introduced. For this purpose the elliptic problem \[ a_1\Delta u + u(b_1 - c_1u- d_1v)=0,\ \varepsilon_1\Delta u+v(b_2-c_2v-d_2u)=0\quad\text{in }\Omega, \ u=v=0\quad\text{on }\partial \Omega, \] where \(0<\varepsilon \ll a_1\) is considered. The bifurcation of nonnegative stationary solutions is studied together with their stability character and their dependence on the ``space'' variables.
reviewed by: Willy G. Engel
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35B32
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 35A01
Mathematics Subject Classification ID: 35K20
zbMATH DE Number: 3846078
zbMATH Keywords: stationary solutions
zbMATH Keywords: competition system
zbMATH Keywords: diffusion
zbMATH Keywords: quasilinear parabolic system
zbMATH Keywords: initial boundary value problem
zbMATH Keywords: existence
zbMATH Keywords: uniqueness
zbMATH Keywords: asymptotic stability
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4171/zaa/59
OpenAlex ID: W2567010963
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description: scientific article; zbMATH DE number 3846078