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Existence of KPP Type Fronts in Space-Time Periodic Shear Flows and a Study of Minimal Speeds Based on Variational Principle

James Nolen·Jack Xin
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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.

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James Nolen, Jack Xin. 2004-07-21. Existence of KPP Type Fronts in Space-Time Periodic Shear Flows and a Study of Minimal Speeds Based on Variational Principle. https://arxiv.org/abs/math/0407366

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