Reaction-diffusion problems with non-Fredholm operators
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.
scientific article; zbMATH DE number 5572856