Stability of localized structures in non-local reaction-diffusion equations

The stability of non-homogeneous, steady state solutions of a scalar, non-local reaction-diffusion equation is considered. Sufficient conditions are provided that guarantee that the relevant linear operator possesses a countable infinity of discrete eigenvalues. These eigenvalues are shown to interlace the eigenvalues of a related local Sturm-Liouville operator. An oscillation theorem for the corresponding non-local eigenfunctions also is established. These results are applied to assess the stability of n-pulse solutions of a model which describes hot spot formation in a microwave heated ceramic fiber. Each n-pulse solution contains n spatially localized regions of elevated temperature. It is shown that the 1-pulse solution is metastable in that the principal eigenvalue of the corresponding linear operator is exponentially small. For n > 2, all solutions are unstable with corresponding principal eigenvalues bounded away from the origin.