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2008 Reaction-diffusion problems with non-Fredholm operators
A. Ducrot, M. Marion, V. Volpert
Adv. Differential Equations 13(11-12): 1151-1192 (2008).

Abstract

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 non-linear 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.

Citation

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A. Ducrot. M. Marion. V. Volpert. "Reaction-diffusion problems with non-Fredholm operators." Adv. Differential Equations 13 (11-12) 1151 - 1192, 2008.

Information

Published: 2008
First available in Project Euclid: 18 December 2012

zbMATH: 1180.35086
MathSciNet: MR2483134

Subjects:
Primary: 35J57
Secondary: 35A01, 35B32, 35J91, 45K05, 47J05, 47N20

Rights: Copyright © 2008 Khayyam Publishing, Inc.

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Vol.13 • No. 11-12 • 2008
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