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

TL;DR

The paper proves the existence of $KPP$ traveling fronts in space-time periodic, mean-zero incompressible advection and derives a variational formula for their minimal propagation speeds. The existence is shown via a limiting process from a regularized combustion-type front equation with ignition cut-off; compactness comes from front monotonicity and the degenerate parabolic structure, while a dynamic argument using the parabolic maximum principle justifies that the fronts propagate at minimal speeds. A central result analyzes the principal eigenvalue $\mu(\lambda)$ of a time-periodic, non-self-adjoint operator, establishing its convexity and a variational characterization $\mu = \inf_{\psi\in E^+}\sup_{(x,t)} \frac{L\psi}{\psi}$, yielding a unique global minimum of $\lambda \mapsto \mu(\lambda)/\lambda$ that determines the minimal speed. The conclusions confirm the existence of $KPP$ fronts with speeds given by this variational principle, extend to general positive nonlinearities, and point to future work on how advection intensity affects the speeds.

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-diffusion-advection equation. The dynamic method does not rely on existence of traveling fronts.

Theorems & Definitions (22)

• Definition 1.1
• Theorem 2.1
• Proposition 2.2
• Theorem 2.3
• Proposition 2.4
• proof
• Theorem 2.5
• proof
• Proposition 2.6
• proof