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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

Analysis of PDEs mathscidoc:1912.43843

arXiv preprint math/0407366, 2004.7
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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@inproceedings{james2004existence,
  title={Existence of KPP type fronts in space-time periodic shear flows and a study of minimal speeds based on variational principle},
  author={James Nolen, and Jack Xin},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224210137749081407},
  booktitle={arXiv preprint math/0407366},
  year={2004},
}
James Nolen, and Jack Xin. Existence of KPP type fronts in space-time periodic shear flows and a study of minimal speeds based on variational principle. 2004. In arXiv preprint math/0407366. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224210137749081407.
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