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An extension of the principle of spatial averaging for inertial manifolds

Part of: Parabolic equations and systems Qualitative theory Spectral theory and eigenvalue problems

Published online by Cambridge University Press:  09 April 2009

Hyukjin Kwean
Hyukjin Kwean
Affiliation:
Department of Mathematics Education, College of Education, Korea University, Sungbuk-Ku Anam-Dong, 136-701 Seoul, Korea e-mail: [email protected]
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Abstract

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In this paper we extend a theorem of Mallet-Paret and Sell for the existence of an inertial manifold for a scalar-valued reaction diffusion equation to new physical domains ωn ⊂ Rn, n = 2,3. For their result the Principle of Spatial Averaging (PSA), which certain domains may possess, plays a key role for the existence of an inertial manifold. Instead of the PSA, we define a weaker PSA and prove that the domains φn with appropriate boundary conditions for the Laplace operator, δ, satisfy a weaker PSA. This weaker PSA is enough to ensure the existence of an inertial manifold for a specific class of scalar-valued reaction diffusion equations on each domain ωn under suitable conditions.

Keywords

dissipative systemsinertial manifoldreaction diffusion equationprinciple of spatial averaging

MSC classification

Secondary: 35P20: Asymptotic distribution of eigenvalues and eigenfunctions 34C29: Averaging method 35K57: Reaction-diffusion equations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 66 , Issue 1 , February 1999 , pp. 125 - 142
Copyright
Copyright © Australian Mathematical Society 1999

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