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Discrete and Continuous Dynamical Systems - Series A (DCDS-A)
 

Existence of KPP type fronts in space-time periodic shear flows and a study of minimal speeds based on variational principle

Pages: 1217 - 1234, Volume 13, Issue 5, December 2005      doi:10.3934/dcds.2005.13.1217

 
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James Nolen - Department of Mathematics, University of Texas at Austin, Austin, TX 78712-0257, United States (email)
Jack Xin - Department of Mathematics, University of Texas at Austin, Austin, TX 78712-0257, United States (email)

Abstract: We prove the existence of reaction-diffusion traveling fronts in mean zero space-time periodic shear flows for nonnegative reactions including the classical KPP (Kolmogorov-Petrovsky-Piskunov) nonlinearity. For the KPP nonlinearity, the minimal front speed is characterized by a variational principle involving the principal eigenvalue of a space-time periodic parabolic operator. Analysis of the variational principle shows that adding a mean-zero space time periodic shear flow to an existing mean zero space-periodic shear flow leads to speed enhancement. Computation of KPP minimal speeds is performed based on the variational principle and a spectrally accurate discretization of the principal eigenvalue problem. It shows that the enhancement is monotone decreasing in temporal shear frequency, and that the total enhancement from pure reaction-diffusion obeys quadratic and linear laws at small and large shear amplitudes.

Keywords:  traveling waves, fronts, heterogeneous media.
Mathematics Subject Classification:  Primary: 35K55, 35K57; Secondary: 41A60, 65D99.

Received: July 2004;      Revised: February 2005;      Available Online: September 2005.