Reaction-diffusion problems with non-Fredholm operators
By Arnaud Ducrot, M. Marion, Vitaly Volpert. Advances in Differential Equations, 2008
90 citations recorded by the source corpus.
The paper is devoted to the study of a multi-dimensional semi-linear elliptic system of equations in an unbounded cylinder with a linear dependence of the components of the non-linearity vector. Problems of this type describe reaction-diffusion waves with the Lewis number different from 1. Due to this property of non-linearity, the corresponding operator does not possess the Fredholm property. Therefore the usual solvability conditions and the conventional methods of nonlinear analysis cannot be directly applied. We reduce the elliptic problem to an integro-differential system of equations and show how to apply the implicit function theorem to it. It allows us to prove existence of waves for the Lewis number different from 1 and sufficiently close to it. Next we prove the Fredholm property of integro-differential operators, their properness, and construct the topological degree. The latter is used to study bifurcations of solutions.
Subfields: Applied Mathematics, Computational Theory and Mathematics, Numerical Analysis
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In Applied Mathematics
- V. Vougalter
- Vitaly Volpert
- C. Camere
- E. Witten
- Weiping Zhang
- Nicolas Ginoux
- I. Shovkovy
- V. A. Miransky
In Computational Theory and Mathematics
- V. Vougalter
- Vitaly Volpert
- C. Camere
- E. Witten
- Weiping Zhang
- Nicolas Ginoux
- I. Shovkovy
- V. A. Miransky
In Numerical Analysis
Citations in the corpus
- Solvability in the sense of sequences to some non-Fredholm operators (2013, background): “Let us note that elliptic problems involving non-Fredholm operators were studied extensively in recent years (see [15], [16]– [22], also [5]) along with their potential applications to the theory of reaction-diffusion equations (see [7], [8]).”
- On the Solvability Conditions for a Linearized Cahn-Hilliard Equation (2011, unknown): “[5], [7], [14] ).”