Existence and convergence to a propagating terrace in one-dimensional reaction-diffusion equations
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- by Arnaud Ducrot, Thomas Giletti and Hiroshi Matano PDF
- Trans. Amer. Math. Soc. 366 (2014), 5541-5566 Request permission
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.References
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Additional Information
- Arnaud Ducrot
- Affiliation: UMR CNRS 5251, Université de Bordeaux, 33000 Bordeaux, France
- MR Author ID: 724386
- 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
- Received by editor(s): March 12, 2012
- Received by editor(s) in revised form: February 4, 2013
- Published electronically: February 10, 2014
- © Copyright 2014 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 366 (2014), 5541-5566
- MSC (2010): Primary 35K55, 35C07, 35B08, 35B40
- DOI: https://doi.org/10.1090/S0002-9947-2014-06105-9
- MathSciNet review: 3240934