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We study the radially symmetric high dimensional Fisher-KPP nonlocaldiffusion equation with free boundary, and reveal some fundamental differencesfrom its one dimensional version considered in \cite{cdjfa} recently.Technically, this high dimensional problem is much more difficult to treatsince it involves two kernel functions which arise from the original kernelfunction $J(|x|)$ in rather implicit ways. By introducing new techniques, weare able to determine the long-time dynamics of the model, including firstlyfinding the threshold condition on the kernel function that governs the onsetof accelerated spreading, and the determination of the spreading speed when itis finite. Moreover, for two important classes of kernel functions, sharpestimates of the spreading profile are obtained. More precisely, for kernelfunctions with compact support, we show that logarithmic shifting occurs fromthe finite wave speed propagation, which is strikingly different from the onedimension 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 acceleratedspreading is possible. These sharp estimates are obtained by constructingsubtle upper and lower solutions, based on careful analysis of the involvedkernel functions.

The high dimensional Fisher-KPP nonlocal diffusion equation with free boundary and radial symmetry