Propagation acceleration in reaction diffusion equations with anomalous diffusions
Résumé
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.
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