Axi-Symmetric, Time-Periodic Solutions of Reaction-Diffusion Equations

In this note we establish the existence of outgoing, target pattern solutions to a system of reaction-diffusion equations introduced by Kopell and Howard and customarily referred to as the “$\lambda$-$\omega $” system. The result is established by reducing the given problem to a nonlinear integral equation and obtaining estimates which guarantee the integral operator is a contracting map on an appropriately chosen function space. A byproduct of this analysis is that the small diffusion perturbation theory for this equation is valid.