A multiple scale pattern formation cascade in reaction-diffusion systems of activator-inhibitor type

Abstract

A family of singular limits of reaction-diffusion systems of activator-inhibitor type in which stable stationary sharp-interface patterns may form is investigated. For concreteness, the analysis is performed for the FitzHugh-Nagumo model on a suitably rescaled bounded domain in R , with N ≥ 2. It is proved that when the system is sufficiently close to the limit the dynamics starting from the appropriate smooth initial data breaks down into five distinct stages on well-separated time scales, each of which can be approximated by a suitable reduced problem. The analysis allows to follow fully the progressive refinement of spatiotemporal patterns forming in the systems under consideration and provides a framework for understanding the pattern formation scenarios in a large class of physical, chemical, and biological systems modeled by the considered class of reaction-diffusion equations. ∗CMI Université d’Aix-Marseille, 39 rue Frédéric Joliot-Curie 13453 Marseille cedex 13, France †Laboratoire de Mathématiques, CNRS and University Paris-Sud Paris-Saclay, 91405 Orsay Cedex, France ‡Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102, USA