Front propagation in periodic excitable media

Abstract

This paper is devoted to the study of pulsating traveling fronts for reactiondiffusion-advection equations in a general class of periodic domains with underlying periodic diffusion and velocity fields. Such fronts move in some arbitrarily given direction with an unknown effective speed. The notion of pulsating traveling fronts generalizes that of traveling fronts for planar or shear flows.Various existence, uniqueness, and monotonicity results are proved for two classes of reaction terms. First, for a combustion-type nonlinearity, it is proved that the pulsating traveling front exists and that its speed is unique. Moreover, the front is increasing with respect to the time variable and unique up to translation in time. We also consider one class of monostable nonlinearity that arises either in combustion or biological models. Then, the set of possible speeds is a semiinfinite interval, closed and bounded from below. For each possible speed, there exists a pulsating traveling front that is increasing in time. This result extends the classical Kolmogorov-Petrovsky-Piskunov case. Our study covers in particular the case of flows in all of space with periodic advections such as periodic shear flows or a periodic array of vortical cells. These results are also obtained for cylinders with oscillating boundaries or domains with a periodic array of holes.

Stability of travelling waves in a model for conical flames in two space dimensions

“…Similar methods were used in [35] and [3] to get some monotonicity results for the solutions of some semilinear parabolic equations in various domains. Theorem 1.3 especially implies the following Theorem 1.5 (Convergence of a subsequence to a travelling wave) Let φ be a solution of (1.1-1.2) for α ∈ (0, π/2] with assumptions (1.4) on f .…”

The speed of propagation for KPP type problems. II: General domains

“…Namely, if u 0 is a nonnegative continuous function in R N with compact support and u 0 ≡ 0, then the solution u(t, x) of (1.1) with initial condition u 0 at time t = 0 spreads with the speed c * in all directions for large times: as t → +∞, max |x|≤ct |u(t, x) − 1| → 0 for each c ∈ [0, c * ) and max |x|≥ct u(t, x) → 0 for each c > c * . In Part I of [7] and in an earlier paper [4], we introduced a general heterogeneous periodic framework extending (1.1). The types of equations which were considered there were (1.3) u t − ∇ · (A(x)∇u) + q(x) · ∇u = f (x, u) in Ω, ν · A∇u = 0 on ∂Ω,…”

“…The open set Ω is clearly strongly unbounded in the direction e. 4 But such a domain does not satisfy the assumptions of Theorem 1.6 (more precisely, Ω does not satisfy Hypothesis H y,y , for any y and y such that…”

Propagation of Reactions in Inhomogeneous Media

“…Traveling fronts are front-like entire (with t 0 = −∞) solutions of (1.1) moving with a constant speed c in a unit direction e ∈ R d , of the form u(t, x) = U (x · e − ct) with lim s→−∞ U (s) = 1 and lim s→∞ U (s) = 0. Pulsating fronts, first introduced by Shigesada, Kawasaki, Teramoto [23] and proved to exist for general periodic f as above by Xin [27] and Berestycki, Hamel [4], are similar but u(t, x) = U (x · e − ct, x) and U is periodic in the second argument. The minimal of the speeds for which such a front exists for a given unit e ∈ R d is then precisely c e , and we also have s e = inf e ·e>0 [c e /(e · e)].…”

“…Instead of a more comprehensive discussion, we refer to [4,26] and the excellent reviews by Berestycki [3] and Xin [28]. Unsurprisingly, the picture becomes less satisfactory for non-periodic reactions, particularly in the several spatial dimensions case d ≥ 2.…”