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

Abstract

<abstract><p>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 <sup>[<xref ref-type="bibr" rid="b18">18</xref>]</sup>, 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 <sup>[<xref ref-type="bibr" rid="b19">19</xref>]</sup>, 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 <sup>[<xref ref-type="bibr" rid="b19">19</xref>]</sup> become particularly simple in dimension $ 3 $.</p></abstract>