Advances in Differential Equations

Reaction-diffusion problems with non-Fredholm operators

A. Ducrot, M. Marion, and V. Volpert

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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.

Article information

Adv. Differential Equations Volume 13, Number 11-12 (2008), 1151-1192.

First available in Project Euclid: 18 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J57: Boundary value problems for second-order elliptic systems
Secondary: 35A01: Existence problems: global existence, local existence, non-existence 35B32: Bifurcation [See also 37Gxx, 37K50] 35J91: Semilinear elliptic equations with Laplacian, bi-Laplacian or poly- Laplacian 45K05: Integro-partial differential equations [See also 34K30, 35R09, 35R10, 47G20] 47J05: Equations involving nonlinear operators (general) [See also 47H10, 47J25] 47N20: Applications to differential and integral equations


Ducrot, A.; Marion, M.; Volpert, V. Reaction-diffusion problems with non-Fredholm operators. Adv. Differential Equations 13 (2008), no. 11-12, 1151--1192.

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