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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
MathSciNet review: 3080177
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Abstract | References | Similar Articles | Additional Information

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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  • 1. H. Berestycki, F. Hamel, Front propagation in periodic excitable media, Comm. Pure Appl. Math. 55 (2002), pp. 949-1032. MR 1900178 (2003d:35139)
  • 2. H. Berestycki, F. Hamel, G. Nadin, Asymptotic spreading in heterogeneous diffusive excitable media, Journal of Functional Analysis 255 (2008), pp. 2146-2189. MR 2473253 (2009k:35136)
  • 3. H. Berestycki, F. Hamel, N. Nadirashvili, The Speed of Propagation for KPP Type Problems (Periodic Framework), J. Eur. Math. Soc. 7 (2005), pp. 173-213. MR 2127993 (2005k:35186)
  • 4. M. El Smaily, Pulsating travelling fronts: Asymptotics and homogenization regimes, European J. Appl. Math. 19 (2008), pp. 393-434. MR 2431698 (2009j:35147)
  • 5. M. El Smaily, F. Hamel, L. Roques, Homogenization and influence of fragmentation in a biological invasion model, Discrete Contin. Dyn. Syst. 25 (2009), pp. 321-342. MR 2525180 (2010j:35040)
  • 6. M. El Smaily, S. Kirsch, The speed of propagation for KPP reaction-diffusion equations within large drift, Advances in Differential Equations 16 (2011), Numbers 3-4, pp. 361-400. MR 2767082 (2012a:35154)
  • 7. M. El Smaily, The homogenized equation of a heterogenous reaction-diffusion model involving pulsating traveling fronts, Communications in Mathematical Sciences (CMS) 9 (2011), No. 4, pp. 1113-1128. MR 2901819
  • 8. M. El Smaily, F. Hamel, R. Huang, Two dimensional curved fronts in a periodic shear flow, Nonlinear Analysis: Theory, Methods $ \&$ Applications 74 (2011), pp. 6469-6486. MR 2833431
  • 9. A. Kiselev, A. Zlatoš, Quenching of combustion by shear flows, Duke Math. J. 132 (2006), pp. 49-72. MR 2219254 (2007b:35199)
  • 10. A. N. Kolmogorov, I. G. Petrovsky, N. S. Piskunov, Étude de l'équation de la diffusion avec croissance de la quantité de matiére et son application a un probléme biologique, Bulletin Université d'Etat à Moscou (Bjul. Moskowskogo Gos. Univ.), Série internationale A1 (1937), pp. 1-26.
  • 11. B. Li, H. F. Weinberger, L.A. Lewis, Spreading speeds as slowest wave speeds for cooperative systems, Math. Biosci. 196 (2005), no. 1, pp. 82-98. MR 2156610 (2006g:92069)
  • 12. Xing Liang, Xiaotao Lin, Hiroshi Matano, A variational problem associated with the minimal speed of travelling waves for spatially periodic reaction-diffusion equations, Trans. Amer. Math. Soc. 362 (2010), pp. 5605-5633. MR 2661490 (2011d:35243)
  • 13. A. Majda, P. Souganidis, Large scale front dynamics for turbulent reaction-diffusion equations with separated velocity scales, Nonlinearity 7 (1994), no. 1, pp. 1-30. MR 1260130 (95e:35180)
  • 14. G. Nadin, Some dependence results between the spreading speed and the coefficients of the space-time periodic Fisher-KPP equation, European J. Appl. Math. 22 (2011), no. 2, pp. 169-185. MR 2774781 (2012c:35214)
  • 15. G. Nadin, Pulsating traveling fronts in space-time periodic media, C.R. Acad. Sci. Paris I 346 (2008), pp. 951-956. MR 2449634 (2009i:35182)
  • 16. G. Nadin, Reaction-diffusion equations in space-time periodic media, C.R. Acad. Sci. Paris I 345 (2007), pp. 489-493. MR 2375108 (2008j:35102)
  • 17. N. Shigesada, K. Kawasaki, E. Teramoto, Spatial segregation of interacting species, J. Theoret. Biol. 79 (1979), pp. 83-99. MR 540951 (80e:92038)
  • 18. L. Ryzhik, A. Zlatoš, KPP pulsating front speed-up by flows, Comm. Math. Sci. 5 (2007), pp. 575-593. MR 2352332 (2008h:35200)
  • 19. W. Shen, Variational principle for spreading speeds and generalized propagating speeds in time almost periodic and space periodic KPP models, Trans. Amer. Math. Soc. 362 (2010), no. 10, pp. 5125-5168. MR 2657675 (2011h:35162)
  • 20. H. F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol. 45 (2002), no. 6, pp. 511-548. MR 1943224 (2004b:92043a)
  • 21. J. X. Xin, Front propagation in heterogeneous media, SIAM Review 42 (2000), pp. 161-230. MR 1778352 (2001i:35184)
  • 22. A. Zlatoš, Sharp asymptotics for KPP pulsating front speed-up and diffusion enhancement by flows, Arch. Ration. Mech. Anal. 195 (2010), pp. 441-453. MR 2592283 (2011b:35248)

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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

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.

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