We study reaction-di¤usion systems of propagator-controller type in the one-dimensional unit interval. When propagator di¤uses slowly, we establish the existence of transition layer equilibria by using singular perturbation expansions and a Lyapunov-Schmidt reduction method. Our approach to the existence also enables us to simultaneously obtain a stability criterion for the layer equilibria.
"A Lyapunov-Schmidt method for transition layers in reaction-diffusion systems." Hiroshima Math. J. 35 (2) 205 - 249, July 2005. https://doi.org/10.32917/hmj/1150998273