A geometric approach to singularly perturbed nonlocal reaction-diffusion equations

Document Type

Article

Publication Date

1-1-2000

Abstract

In the context of a microwave heating problem, a geometric method to construct a spatially localized, 1-pulse steady-state solution of a singularly perturbed, nonlocal reaction-diffusion equation is introduced. The 1-pulse is shown to lie in the transverse intersection of relevant invariant manifolds. The transverse intersection encodes a consistency condition that all solutions of nonlocal equations must satisfy. An oscillation theorem for eigenfunctions of nonlocal operators is established. The theorem is used to prove that the linear operator associated with the 1-pulse solution possesses an exponentially small principal eigenvalue. The existence and instability of n-pulse solutions is also proved. A further application of the theory to the Gierer-Meinhardt equations is provided. © 2000 Society for Industrial and Applied Mathematics.

Identifier

0033474943 (Scopus)

Publication Title

SIAM Journal on Mathematical Analysis

External Full Text Location

https://doi.org/10.1137/S0036141098342556

ISSN

00361410

First Page

431

Last Page

454

Issue

2

Volume

31

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