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

A computational study of residual KPP front speeds in time-periodic cellular flows in the small diffusion limit

“…If the initial data for u is nonnegative and compactly supported, the large time behavior of u is an outward propagating front, with speed c * = c * ( e) in the direction e = 1, 0 . The variational principle of c * is [18]:…”

Existence of Positive Solutions of Fisher-KPP Equations in Locally Spatially Variational Habitat with Hybrid Dispersal

“…We refer to (Aronson & Weinberger, 1957;Aronson & Weinberger, 1978;Berestycki, Hamel, & Nadirashvili, 2010;Kametaka, 1976;Liang & Zhao, 2007;Liang, Yi, & Zhao, 2006;Sattinger, 1976;Uchiyama, 1978;Weinberger, 1982;etc). for the study of (1.2) in the case that f (x, u) is independent of x and refer to (Berestycki, Hamel, & Nadirashvili, 2005;Berestycki, Hamel, & Roques, 2005;Freidlin & Gärtner, 1979;Hamel, 2008;Hudson & Zinner, 1995;Nadin, 2009;Nolen, Rudd, & Xin, 2005;Weinberger, 2002;etc). for the study of (1.2) in the case that f (x, u) is periodic in x; refer to (Coville & Dupaigne, 2005;Coville, Dávila, & Martínez, 2008;Li, Sun, & Wang, 2010;etc).…”

A Variational Principle for KPP Front Speeds in Temporally Random Shear Flows

“…Our proofs are built on his, with additional ingredients to handle both the time-dependence and the stochastic nature of the field B. For example, in the periodic case, µ(λ) is the principal eigenvalue of a periodic-parabolic operator [28,26], and perturbation theory [17] implies that µ(λ) is differentiable in λ. It then follows from Theorem 7.1.1 and Theorem 7.1.2 of [14] that the random variables η t z (t) satisfy a large deviation principle with convex rate function S(c) given by (4.3).…”

“…For compactly supported initial data bounded between 0 and 1, solutions of (1.1) develop into propagating fronts separating the cylindrical domain into a region where u ≈ 1 and the rest where u ≈ 0, which correspond to burned (hot) and unburned (cold) states in combustion. In case B is periodic in z and t, KPP type front dynamics and speeds have been recently studied for both shear and more general incompressible flows [20,22,23,27,28,26]. Exact traveling front solutions exist [27,28,26], extending those in spatially periodic media, [4,5,36,35], see also [3] and [37] for reviews.…”