The high dimensional Fisher-KPP nonlocal diffusion equation with free boundary and radial symmetry, part 2: Sharp estimates

Abstract

This is the second part of a two-part series devoted to an in depth understanding of the dynamical behaviour of the radially symmetric Fisher-KPP nonlocal diffusion equation with free boundary |x|=h(t). In Part 1 [19], we have shown that the long-time dynamics of this problem is characterised by a spreading-vanishing dichotomy, and there exists a threshold condition on the diffusion kernel J(|x|) such that the spreading speed is ∞ when this condition is not satisfied, and when it is satisfied, the finite spreading speed c0 is determined by an associated semi-wave problem established in [15]. In Part 2 here, we obtain more precise description of the spreading profile by focusing on some natural classes of kernel functions, including those satisfying J(|x|)∼|x|−β for |x|≫1 in RN. Our results for such kernels reveal a striking difference of behaviour from the pattern exhibited in the one dimension case [18] when β crosses the value N+2. More precisely, (a) when β∈(N,N+1], we show that for t≫1, h(t)∼t1/(β−N) if β∈(N,N+1), and h(t)∼tln⁡t if β=N+1, which is of the same pattern as in dimension one, namely we recover the result in [18] by letting N=1 in the above statements; (b) when β∈(N+1,N+2], the front has a finite spreading speed c0=c0(β) in the sense that limt→∞⁡h(t)/t=c0, and our results here on the order of shift c0t−h(t) are again of the same pattern as in dimension one; (c) when β>N+2, the front still has a finite spreading speed c0, but a significant change happens to the order of shift c0t−h(t) between N≥2 and N=1: for t≫1, c0t−h(t)∼ln⁡t when N≥2, but c0t−h(t)∼1 when N=1.