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CONVERGENCE OF SOLUTIONS TO SOME ELLIPTIC EQUATIONS IN BOUNDED NEUMANN THIN DOMAINS

Part of: Partial differential equations Parabolic equations and systems General mathematical topics and methods in quantum theory

Published online by Cambridge University Press:  20 November 2015

CÉSAR R. DE OLIVEIRA  and
ALESSANDRA A. VERRI
CÉSAR R. DE OLIVEIRA*
Affiliation:
Departamento de Matemática, UFSCar, São Carlos, SP 13560-970, Brazil email oliveira@dm.ufscar.br
ALESSANDRA A. VERRI
Affiliation:
Departamento de Matemática, UFSCar, São Carlos, SP 13560-970, Brazil email alessandraverri@dm.ufscar.br
*
✉oliveira@dm.ufscar.br
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Abstract

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Consider the (elliptic) stationary nonlinear reaction–diffusion equation in a sequence of bounded Neumann tubes in a space that is squeezed to a reference curve. It is supposed that the forcing term is square integrable and that the nonlinear one satisfies some growth and dissipative conditions. A norm convergence of the resolvents of the operators associated with the linear terms of such equations is proven, and this fact is used to provide new and simpler proofs of the asymptotic behaviour of the solutions to the full nonlinear equations (previously known in similar singular problems).

Keywords

thin tubesresolvent convergencereduction of dimension

MSC classification

Primary: 35B25: Singular perturbations
Secondary: 35K57: Reaction-diffusion equations 81Q15: Perturbation theories for operators and differential equations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 100 , Issue 2 , April 2016 , pp. 252 - 271
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

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