Communications in Mathematical Sciences

Volume 17 (2019)

Number 2

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

Pages: 565 – 586



Alexis Léculier (Institut de Mathématiques de Toulouse, Université Paul Sabatier, Toulouse, France)


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.


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

2010 Mathematics Subject Classification

35B40, 35K57, 35Q92

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