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

Abstract

We prove the existence of reaction-diffusion traveling fronts inmean zero space-time periodic shear flows fornonnegative reactions including theclassical KPP (Kolmogorov-Petrovsky-Piskunov) nonlinearity.For the KPP nonlinearity, the minimal front speed is characterizedby a variational principle involving the principal eigenvalue ofa space-time periodic parabolic operator.Analysis of the variational principle showsthat adding a mean-zero space time periodic shear flow toan existing mean zero space-periodic shear flow leads tospeed enhancement. Computation of KPP minimal speeds is performedbased on the variational principleand a spectrally accurate discretizationof the principal eigenvalue problem. It shows that the enhancementis monotone decreasing in temporal shear frequency, and thatthe total enhancement from pure reaction-diffusionobeys quadratic and linear laws at small and largeshear amplitudes.