Dynamics of Partial Differential Equations

Volume 2 (2005)

Number 1

Existence of KPP fronts in spatially-temporally periodic advection and variational principle for propagation speeds

Pages: 1 – 24

DOI: http://dx.doi.org/10.4310/DPDE.2005.v2.n1.a1

Authors

James Nolen (Department of Mathematics, University of Texas at Austin)

Matthew Rudd (Department of Mathematics, University of Texas at Austin)

Jack Xin (Department of Mathematics and Institute of Computational Engineering and Sciences, University of Texas at Austin)

Abstract

We prove the existence of Kolmogorov-Petrovsky-Piskunov (KPP) type traveling fronts in space-time periodic and mean zero incompressible advection, and establish a variational (minimization) formula for the minimal speeds. We approach the existence by considering limit of a sequence of front solutions to a regularized traveling front equation where the nonlinearity is combustion type with ignition cut-off. The limiting front equation is degenerate parabolic and does not permit strong solutions, however, the necessary compactness follows from monotonicity of fronts and degenerate regularity. We apply a dynamic argument to justify that the constructed KPP traveling fronts propagate at minimal speeds, and derive the speed variational formula. The dynamic method avoids the degeneracy in traveling front equations, and utilizes the parabolic maximum principle of the governing reaction-diffusionadvection equation. The dynamic method does not rely on existence of traveling fronts.

Keywords

Kolmogorov-Petrovsky-Piskunov (KPP) type traveling front, reaction-diffusion-advection (RDA) equation, variational (minimization) formula

2010 Mathematics Subject Classification

Primary 35-xx, 76-xx. Secondary 37-xx.

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