Propagation phenomena for time heterogeneous KPP reaction–diffusion equations

Abstract

We investigate in this paper propagation phenomena for the heterogeneous reaction-diffusion equation∂ t u - Δ u = f (t, u), x ∈ R N, t ∈ R, where f = f (t, u) is a KPP monostable nonlinearity which depends in a general way on t ∈ R. A typical f which satisfies our hypotheses is f (t, u) = μ (t) u (1 - u), with μ ∈ L ∞ (R) such that ess inf t ∈ R μ (t) > 0. We first prove the existence of generalized transition waves (recently defined in Berestycki and Hamel (2007) ) for a given class of speeds. As an application of this result, we obtain the existence of random transition waves when f is a random stationary ergodic function with respect to t ∈ R. Lastly, we prove some spreading properties for the solution of the Cauchy problem

Parabolic–elliptic chemotaxis model with space–time-dependent logistic sources on ℝN. I. Persistence and asymptotic spreading

“…A huge amount of research has been carried out toward various extensions of traveling wave solutions and take-over properties of (1.4) to general time and space independent as well as time and/or space dependent Fisher-KPP type equations. See, for example, [3], [4], [9], [15], [21], [47], [53], etc., for the extension to general time and space independent Fisher-KPP type equations; see [5,6,11,20,26,27,28,31,38,39,44,54,55], and references therein for the extension to time and/or space periodic Fisher-KPP type equations; and see [7,8,16,19,30,32,33,34,35,37,49,50,51,52,57,58], and references therein for the extension to quite general time and/or space dependent Fisher-KPP type equations. It should be pointed out that the so called periodic traveling wave solutions or pulsating traveling fronts to time and/or space periodic reaction diffusion equations are natural extension of the notion of traveling wave solutions in the classical sense, and that the so called transition fronts or generalized traveling waves to general time and/or space dependent reaction equations are the natural extension of the notion of traveling wave solutions in the classical sense (see [7,8] for the introduction of the notion of transition fro...…”

Critical travelling waves for general heterogeneous one-dimensional reaction–diffusion equations

“…A similar quantity has been introduced by Rossi and the author in [22]. They proved that, in the framework of time-dependent monostable equations, there exists an explicit treshold c * such that spatial transition wave with least mean speed c exist for all c > c * and do not exist if c < c * .…”

“…The existence of spatial transition waves for monostable time-dependent equations have been proved when the coefficients are assumed to be uniquely ergodic by Shen [27] and in the general framework by Rossi and the author [22]. In these two papers, the nonlinearity is assumed to be KPP, that is, it is C 1 in u = 0 and f (t, u) ≤ f ′ u (t, 0)u for all (t, u) ∈ R × [0, 1].…”

“…Hence, in all the cases where spatial transition waves are known to exist and to be unique [19,22,26], these solutions are indeed generalized travelling waves in the sense of Matano.…”

Generalized fronts in reaction-diffusion equations with bistable nonlinearity

“…In a very recent paper [22], Shen extends the results of time and space periodic monostable equations to time recurrent and space periodic ones, and obtained the transition fronts. Nadin and Rossi [23] investigated the propagation phenomena for the time heterogeneous media. Namely, they assume that f = μ(t)f 0 (u), with μ ∈ L ∞ (R) and inf t∈R μ(t) > 0.…”