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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Existence and convergence to a propagating terrace in one-dimensional reaction-diffusion equations


Authors: Arnaud Ducrot, Thomas Giletti and Hiroshi Matano
Journal: Trans. Amer. Math. Soc. 366 (2014), 5541-5566
MSC (2010): Primary 35K55, 35C07, 35B08, 35B40
Published electronically: February 10, 2014
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Abstract: We consider one-dimensional reaction-diffusion equations for a large class of spatially periodic nonlinearities - including multi-stable ones - and study the asymptotic behavior of solutions with Heaviside type initial data. Our analysis reveals some new dynamics where the profile of the propagation is not characterized by a single front, but by a layer of several fronts which we call a terrace. Existence and convergence to such a terrace is proven by using an intersection number argument, without much relying on standard linear analysis. Hence, on top of the peculiar phenomenon of propagation that our work highlights, several corollaries will follow on the existence and convergence to pulsating traveling fronts even for highly degenerate nonlinearities that have not been treated before.


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Additional Information

Arnaud Ducrot
Affiliation: UMR CNRS 5251, Université de Bordeaux, 33000 Bordeaux, France

Thomas Giletti
Affiliation: UMR 6632 LATP, Université Aix-Marseille, Faculté des Sciences et Techniques, 13397 Marseille, France
Address at time of publication: UMR 7502 IECL, Université de Lorraine, 54506 Vandoeuvre-lès-Nancy, France

Hiroshi Matano
Affiliation: Graduate School of Mathematical Sciences, University of Tokyo, Komaba, Tokyo 153-8914, Japan

DOI: http://dx.doi.org/10.1090/S0002-9947-2014-06105-9
PII: S 0002-9947(2014)06105-9
Keywords: Multi-stable reaction-diffusion equation, periodic environment, long time behavior, propagating terrace, zero-number argument
Received by editor(s): March 12, 2012
Received by editor(s) in revised form: February 4, 2013
Published electronically: February 10, 2014
Article copyright: © Copyright 2014 American Mathematical Society