  We study the radially symmetric high dimensional Fisher-KPP nonlocal
diffusion equation with free boundary, and reveal some fundamental differences
from its one dimensional version considered in \cite{cdjfa} recently.
Technically, this high dimensional problem is much more difficult to treat
since it involves two kernel functions which arise from the original kernel
function $J(|x|)$ in rather implicit ways. By introducing new techniques, we
are able to determine the long-time dynamics of the model, including firstly
finding the threshold condition on the kernel function that governs the onset
of accelerated spreading, and the determination of the spreading speed when it
is finite. Moreover, for two important classes of kernel functions, sharp
estimates of the spreading profile are obtained. More precisely, for kernel
functions with compact support, we show that logarithmic shifting occurs from
the finite wave speed propagation, which is strikingly different from the one
dimension case; for kernel functions $J(|x|)$ behaving like $|x|^{-\beta}$ for
$x\in\R^N$ near infinity, we obtain the rate of accelerated spreading when
$\beta\in (N, N+1]$, which is the exact range of $\beta$ where accelerated
spreading is possible. These sharp estimates are obtained by constructing
subtle upper and lower solutions, based on careful analysis of the involved
kernel functions.
