A singular limit in a fractional reaction-diffusion equation with periodic coefficients

We provide an asymptotic analysis of a non-local Fisher-KPP-type equation in periodic media and with a non-local stable operator of order $\alpha \in (0,1)$. We perform a long time-long range scaling in order to prove that the stable state invades the unstable state with a speed which is exponential in time.

Keywords

non-local fractional operator, Fisher KPP, asymptotic analysis, exponential speed of propagation, perturbed test function

2010 Mathematics Subject Classification

35B40, 35K57, 35Q92

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The author’s research was funded by the European Research Council under the European Union’s Seventh Framework Program (FP/2007-2013) / ERC Grant Agreement n.321186 - ReaDi - Reaction-Diffusion Equations, Propagation and Modeling; and by the French ANR project MODEVOL ANR-13-JS01-0009.

Received 27 April 2018

Received revised 27 December 2018

Accepted 27 December 2018

Published 8 July 2019