An Asymptotic Analysis of Localized Solutions for Some Reaction‐Diffusion Models in Multidimensional Domains
Michael J. Ward · Studies in Applied Mathematics · 1996
In the limit ∊ → 0, a spike‐layer solution is constructed for the reaction‐diffusion equation urn:x-wiley:00222526:media:sapm1996972103:sapm1996972103-math-0001 where 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.