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

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 $\left|x\right|=h\left(t\right)$. 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 $J\left(\right|x\left|\right)$ such that the spreading speed is ∞ when this condition is not satisfied, and when it is satisfied, the finite spreading speed $c_0$ 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 $J\left(\right|x\left|\right)\sim |x{|}^{-\beta }$ for $\left|x\right|\gg 1$ in $R^N$. 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 $\beta \in (N,N+1]$, we show that for $t\gg 1$, $h\left(t\right)\sim t^{1/(\beta -N)}$ if $\beta \in (N,N+1)$, and $h\left(t\right)\sim t\ln t$ if $\beta =N+1$, 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 $\beta \in (N+1,N+2]$, the front has a finite spreading speed $c_0=c_0\left(\beta \right)$ in the sense that ${\lim }_{t\to \infty }h\left(t\right)/t=c_0$, and our results here on the order of shift $c_{0}t-h\left(t\right)$ are again of the same pattern as in dimension one; (c) when $\beta >N+2$, the front still has a finite spreading speed $c_0$, but a significant change happens to the order of shift $c_{0}t-h\left(t\right)$ between $N\ge 2$ and $N=1$: for $t\gg 1$, $c_{0}t-h\left(t\right)\sim \ln t$ when $N\ge 2$, but $c_{0}t-h\left(t\right)\sim 1$ when $N=1$.