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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 , with . It is shown 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 spatio-temporal 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.
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Marie Henry, Danielle Hilhorst, Cyrill B. Muratov, A multiple scale pattern formation cascade in reaction-diffusion systems of activator-inhibitor type. Interfaces Free Bound. 20 (2018), no. 2, pp. 297–336DOI 10.4171/IFB/403