Reaction-diffusion problems with non-Fredholm operators
type: Item
instance of: scholarly article
title: Reaction-diffusion problems with non-Fredholm operators
zbMATH Open document ID: 1180.35086
published in: Advances in Differential Equations
publication date: 2009-06-30T00:00:00Z
review text: The paper is devoted to the study of reaction-diffusion systems of the form: \[ \begin{aligned} -\Delta u + a(c,y)\frac{\partial u}{\partial x} - k(u,v) = 0 &\quad\text{in \(\Omega\)},\\ -\lambda \Delta v + a(c,y)\frac{\partial v}{\partial x} + k(u,v) = 0 &\quad\text{in \(\Omega\)}, \end{aligned} \] with non-homogeneous boundary conditions. Here \(\Omega = \{(x,y) \in \mathbb R \times\omega\}\) is an unbounded cylinder, where \(\omega\) is a regular bounded domain of \(\mathbb R^d\), \(d =1,2\). The nonlinear terms cancel by adding the previous two equations. Consequently, the underlying elliptic operator does not satisfy Fredholm property. Therefore, the usual methods of nonlinear analysis can not be directly applied to deduce the existence of solutions. To avoid this difficulty, the previous system is reduced to the integro-differential equation: \[ B(v,c,\lambda) : = -\lambda \Delta(v+w) + a(c,y)\frac{\partial (v+w)}{\partial x} + k(H(v) + 1 - v- w,v+w) = 0, \] where \(w \in C^\infty\) takes care of the non-homogeneous boundary conditions and \(H(v)\) is the unique solution to the linear equation: \[ - \Delta h + a(c,y)\frac{\partial h}{\partial x} = (\lambda-1)\Delta v, \] together with boundary conditions. The implicit function theorem applied to \(B\) permits to deduce the existence of solutions for the Lewis number \(\lambda\) different from 1 but close to it. Fredholm properties of integro-differential operators are proved. The topological degree constructed in [\textit{V. Volpert} and \textit{A. Volpert}, Abstr. Appl. Anal. 2003, No.~3, 129--181 (2003; Zbl 1030.35038)] is adapted for elliptic problems in unbounded domains, and it is used to study bifurcation of solutions.
reviewed by: Marlène Frigon
Mathematics Subject Classification ID: 35B32
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 45K05
Mathematics Subject Classification ID: 35J61
Mathematics Subject Classification ID: 47H11
Mathematics Subject Classification ID: 35J57
zbMATH DE Number: 5572856
zbMATH Keywords: Lewis number
zbMATH Keywords: implicit function theorem
zbMATH Keywords: topological degree
zbMATH Keywords: unbounded cylinder
author: Arnaud Ducrot
author: Martine Marion
author: Vitaly Volpert
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5572856