Journal of the European Mathematical Society


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Volume 13, Issue 2, 2011, pp. 345–390
DOI: 10.4171/JEMS/256

Uniqueness and stability properties of monostable pulsating fronts

François Hamel[1] and Lionel Roques[2]

(1) Laboratoire d'Analyse Topologie Probabilités, Université Aix-Marseille III, Avenue Escadrille Normandie-Niemen, 13397, MARSEILLE CEDEX 20, FRANCE
(2) INRA, UR546 Biostatistique et Processus Spatiaux, 33, rue Louis Pasteur, F-84914, AVIGNON, FRANCE

We prove the uniqueness, up to shifts, of pulsating traveling fronts for reaction-diffusion equations in periodic media with Kolmogorov–Petrovskiĭ–Piskunov type nonlinearities. These results provide in particular a complete classification of all KPP pulsating fronts. Furthermore, in the more general case of monostable nonlinearities, we also derive several global stability properties and convergence to pulsating fronts for solutions of the Cauchy problem with front-like initial data. In particular, we prove the stability of KPP pulsating fronts with minimal speed, which is a new result even in the case when the medium is invariant in the direction of propagation.

Keywords: Traveling fronts, periodic media, uniqueness, stability, reaction-diffusion equations, monostable reaction

Hamel F, Roques L. Uniqueness and stability properties of monostable pulsating fronts. J. Eur. Math. Soc. 13 (2011), 345-390. doi: 10.4171/JEMS/256