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An Asymptotic Analysis of Localized Solutions for Some Reaction‐Diffusion Models in Multidimensional Domains
Author(s) -
Ward Michael J.
Publication year - 1996
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1002/sapm1996972103
Subject(s) - eigenfunction , bounded function , mathematics , reaction–diffusion system , domain (mathematical analysis) , limit (mathematics) , mathematical analysis , eigenvalues and eigenvectors , diffusion , exponential growth , regular polygon , diffusion equation , mathematical physics , combinatorics , physics , geometry , quantum mechanics , economy , economics , service (business)
In the limit ∊ → 0, a spike‐layer solution is constructed for the reaction‐diffusion equationwhere b > 0 and D is a bounded convex domain. Here Q ( u ) is such that there exists a unique radially symmetric function u c ( ∊ −1 r ) satisfying ∊ 2 Δ u c + Q ( u c ) = 0 in all of ℛ N , with u c ( ρ ) decaying exponentially at infinity. The spike‐layer solution has the form u ~ u c [ ∊ | x − x 0 |], where the spike‐layer location x 0 ∊ D is to be determined subject to the condition that dist( x 0 , ∂D ) as ∊ → D . The determination of x 0 is shown to be exponentially ill conditioned and asymptotic estimates for the exponentially small eigenvalues and the corresponding eigenfunctions associated with the linearized problem are obtained. These spectral results are used together with a limiting solvability condition to derive an equation for x 0 . For a strictly convex domain, it is shown that there is an x 0 that is located at an O ( ∊ ) distance away from the point in D that is furthest from ∂D . Finally, hot‐spot solutions to Bratu's equation are constructed asymptotically in a singularly perturbed limit.