An Asymptotic Analysis of Localized Solutions for Some Reaction-diffusion Models in Multi-dimensional Domains

In the limit ! 0, a spike-layer solution is constructed for the reaction-diiusion equation 2 4u + Q(u) = 0 ; x 2 D R N ; @ n u + bu = 0 ; x 2 @D ; 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 4u c +Q(u c) = 0 in all of R N , with u c () decaying exponentially at innnity. The spike-layer solution has the form u u c ?1 jx ? x 0 j], where the spike-layer location x 0 2 D is to be found subject to the condition that dist(x 0 ; @D) = O(1) as ! 0. 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 which is furthest from @D. Finally, hot-spot solutions to Bratu's equation are constructed asymptotically in a singularly perturbed limit.