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Transition fronts for the Fisher-KPP equation


Authors: François Hamel and Luca Rossi
Journal: Trans. Amer. Math. Soc. 368 (2016), 8675-8713
MSC (2010): Primary 35B40, 35K57, 35B08, 35C07
DOI: https://doi.org/10.1090/tran/6609
Published electronically: January 26, 2016
MathSciNet review: 3551585
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Abstract: This paper is concerned with transition fronts for reaction-diffusion equations of the Fisher-KPP type. Basic examples of transition fronts connecting the unstable steady state to the stable one are the standard traveling fronts, but the class of transition fronts is much larger and the dynamics of the solutions of such equations is very rich. In the paper, we describe the class of transition fronts and we study their qualitative dynamical properties. In particular, we characterize the set of their admissible asymptotic past and future speeds and their asymptotic profiles and we show that the transition fronts can only accelerate. We also classify the transition fronts in the class of measurable superpositions of standard traveling fronts.


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Additional Information

François Hamel
Affiliation: Aix Marseille Université, CNRS, Centrale Marseille, Institut de Mathématiques de Marseille, UMR 7373, 13453 Marseille, France

Luca Rossi
Affiliation: Dipartimento di Matematica P. e A., Università di Padova, via Trieste 63, 35121 Padova, Italy
Address at time of publication: CNRS, UMR 8557, Centre d’Analyse et de Mathématique Sociales, 75244 Paris Cedex 13, France
Email: luca.rossi@ehess.fr

DOI: https://doi.org/10.1090/tran/6609
Received by editor(s): April 10, 2014
Received by editor(s) in revised form: October 30, 2014
Published electronically: January 26, 2016
Additional Notes: This work was carried out in the framework of the Labex Archimède (ANR-11-LABX-0033) and of the A*MIDEX project (ANR-11-IDEX-0001-02), funded by the “Investissements d’Avenir” French government program managed by the French National Research Agency (ANR). The research leading to these results also received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n.321186 - ReaDi - Reaction-Diffusion Equations, Propagation and Modelling and from Italian GNAMPA-INdAM. Part of this work was carried out during visits by the first author to the Departments of Mathematics of the University of California, Berkeley, of Stanford University and of the Università di Padova, whose hospitality is thankfully acknowledged.
Article copyright: © Copyright 2016 American Mathematical Society