The Fisher-KPP nonlocal diffusion equation with free boundary and radial symmetry in ${\mathbb{R}}^3$

This paper is concerned with the radially symmetric Fisher-KPP nonlocal diffusion equation with free boundary in dimension 3. For arbitrary dimension $N\ge 2$, in ^[18], we have shown that its long-time dynamics is characterised by a spreading-vanishing dichotomy; moreover, we have found a threshold condition on the kernel function that governs the onset of accelerated spreading, and determined the spreading speed when it is finite. In a more recent work ^[19], we have obtained sharp estimates of the spreading rate when the kernel function $J(|x|)$ behaves like $|x{|}^{-\beta }$ as $|x|\to \mathrm{\infty }$ in ${\mathbb{R}}^N$ ( $N\ge 2$). In this paper, we obtain more accurate estimates for the spreading rate when $N=3$, which employs the fact that the formulas relating the involved kernel functions in the proofs of ^[19] become particularly simple in dimension $3$.