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The Dynamics of Localized Solutions of Nonlocal Reaction-Diffusion Equations

Published online by Cambridge University Press:  20 November 2018

Michael J. Ward
Michael J. Ward*
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2
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Abstract

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Many classes of singularly perturbed reaction-diffusion equations possess localized solutions where the gradient of the solution is large only in the vicinity of certain points or interfaces in the domain. The problems of this type that are considered are an interface propagation model from materials science and an activator-inhibitor model of morphogenesis. These two models are formulated as nonlocal partial differential equations. Results concerning the existence of equilibria, their stability, and the dynamical behavior of localized structures in the interior and on the boundary of the domain are surveyed for these two models. By examining the spectrum associated with the linearization of these problems around certain canonical solutions, it is shown that the nonlocal term can lead to the existence of an exponentially small principal eigenvalue for the linearized problem. This eigenvalue is then responsible for an exponentially slow, or metastable, motion of the localized structure.

Keywords

35Q3535C2035K60
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 43 , Issue 4 , 01 December 2000 , pp. 477 - 495
Copyright
Copyright © Canadian Mathematical Society 2000

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