Propagation acceleration in reaction diffusion equations with anomalous diffusions

Abstract

In this paper, we are interested in the properties of solution of the nonlocal equation    ut + (−∆) s u = f (u), t > 0, x ∈ R u(0, x) = u0(x), x ∈ R where 0 ≤ u0 < 1 is a Heaviside type function, ∆ s stands for the fractional Laplacian with s ∈ (0, 1), and f ∈ C([0, 1], R +) is a non negative nonlinearity such that f (0) = f (1) = 0 and f ′ (1) < 0. In this context, it is known that the solution u(t, s) converges locally uniformly to 1 and our aim here is to understand how fast this invasion process occur. When f is a Fisher-KPP type nonlinearity and s ∈ (0, 1), it is known that the level set of the solution u(t, x) moves at an exponential speed whereas when f is of ignition type and s ∈ 1 2 , 1 then the level set of the solution moves at a constant speed. In this article, for general monostable nonlinearities f and any s ∈ (0, 1) we derive generic estimates on the position of the level sets of the solution u(t, x) which then enable us to describe more precisely the behaviour of this invasion process. In particular, we obtain a algebraic generic upper bound on the "speed" of level set highlighting the delicate interplay of s and f in the existence of an exponential acceleration process. When s ∈ 0, 1 2 and f is of ignition type, we also complete the known description of the behaviour of u and give a precise asymptotic of the speed of the level set in this context. Notably, we prove that the level sets accelerate when s ∈ 0, 1 2 and that in the critical case s = 1 2 although no travelling front can exist, the level sets still move asymptotically at a constant speed. These new results are in sharp contrast with the bistable situation where no such acceleration may occur, hightligting therefore the qualitative difference between the two type of nonlinearities.

Sharp exponent of acceleration in general nonlocal equations with a weak Allee effect

“…To give the reader a clear panorama of the rates of invasion occurring in integro-differential models, we may summarise our and previous contributions in Figure 1. Note that the condition on β fits and unifies all related papers [2,25,21].…”

“…In the blue zone, we provide the sharp lower bound, upper bounds being given in [21,3]: 1) . The orange zone is a zone of exponential propagation, after e.g Roquejoffre et al [12], Garnier [24], Bouin et al [11]:…”

“…From there we can then obtain the desire estimate by obtaining the estimate for the fractional case. Let us now explain how to obtain the estimates in the case of the Fractional Laplacian and recall the computation made in [28,21,12]. Let p s (t, x) be the heat kernel for ∆ s , then the solution of (3.6) in this situation is then given by…”

“…As far as (1.1) is concerned, when J ∝ | • | −(1+2s) or integrable with a finite first moment, bounds on the expansion of the level sets of u have been already obtained in [21,3] showing a delicate interplay between the tails of J and the power β. Namely, Coville et al obtained a sharp upper bound of acceleration when J ∝ |•| −(1+2s) or integrable with a finite first moment and assuming that β < 1+ 1 2s−1 : they got an expansion at at most t β 2s(β−1) . It is worth mentioning that the construction made in [3,21] to obtain this bound is very robust and can be straightforwardly adapted to the problem considered here for kernel satisfying (1.1). So to avoid unnecessary computations, we will not duplicate here this construction.…”

“…However, they were unable to provide a matching lower bound: their lower bound was growing faster than linearly, but not fast enough. We do not state the exact exponents they get to avoid misunderstandings while reading this paper, but refer to [21] where they are explicit.…”