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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The non-monotonicity of the KPP speed with respect to diffusion in the presence of a shear flow


Author: Mohammad El Smaily
Journal: Proc. Amer. Math. Soc. 141 (2013), 3553-3563
MSC (2010): Primary 35K57, 92D25, 92D40, 35P15, 35P20, 76F10
Published electronically: June 26, 2013
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Abstract: In this paper, we prove via counterexamples that adding an
advection term of the form Shear flow (whose streamlines are parallel to the direction of propagation) to a reaction-diffusion equation will be enough
heterogeneity to spoil the increasing behavior of the KPP speed of propagation with respect to diffusion. The non-monotonicity of the speed with respect to diffusion will occur even when the reaction term and the diffusion matrices are considered homogeneous (do not depend on space variables). For the sake of completeness, we announce our results in a setting which allows domains with periodic perforations that may or may not be equal to the whole space $ \mathbb{R}^N.$


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Additional Information

Mohammad El Smaily
Affiliation: Department of Mathematical Sciences and Center for Nonlinear Analysis, Carnegie Mellon University, Wean Hall, 5000 Forbes Avenue, Pittsburgh, Pennsylvania 15213 — and — Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais 1049-001 Lisboa, Portugal
Address at time of publication: Department of Mathematics, University of Toronto, 40 St. George Street, Room 6290, Toronto, ON, M5S 2E4, Canada
Email: elsmaily@math.toronto.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2013-11728-4
PII: S 0002-9939(2013)11728-4
Keywords: Traveling fronts, reaction-diffusion, monotonicity with respect to diffusion, KPP minimal speed, shear flows, principal eigenvalue
Received by editor(s): February 28, 2011
Received by editor(s) in revised form: March 9, 2011, November 11, 2011, and January 3, 2012
Published electronically: June 26, 2013
Additional Notes: The author is indebted to the Center for Nonlinear Analysis and Portugal’s Foundation for Science and Technology, “Fundação para a Ciência e a Tecnologia”, for financial and scientific support via the Carnegie Mellon-Portugal Program.
Communicated by: James E. Colliander
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.