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October  2005, 13(5): 1217-1234. doi: 10.3934/dcds.2005.13.1217

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

James Nolen 1, and  Jack Xin 1,

1. 

Department of Mathematics, University of Texas at Austin, Austin, TX 78712-0257, United States, United States

Received  July 2004 Revised  February 2005 Published  September 2005

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: fronts, heterogeneous media., traveling waves.
Mathematics Subject Classification: Primary: 35K55, 35K57; Secondary: 41A60, 65D9.
Citation: James Nolen, Jack Xin. Existence of KPP type fronts in space-time periodic shear flows and a study of minimal speeds based on variational principle. Discrete & Continuous Dynamical Systems - A, 2005, 13 (5) : 1217-1234. doi: 10.3934/dcds.2005.13.1217
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