Abstract
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-diusion 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.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
