A geometric approach to singularly perturbed nonlocal reaction-diffusion equations

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.