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\geq 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\infty $ in $ {\mathbb R}^N $ ($ N\geq 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 $.