The Fisher-KPP nonlocal diffusion equation with free boundary and radial symmetry in R3

Abstract

<p>This paper is concerned with the radially symmetric Fisher-KPP nonlocal diffusion equation with free boundary in dimension 3. For arbitrary dimension <i>N</i> ≥ 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|^<i>−β</i> as |x| → ∞ in R^<i>N</i> (<i>N</i> ≥ 2). In this paper, we obtain more accurate estimates for the spreading rate when <i>N</i> = 3, which employs the fact that the formulas relating the involved kernel functions in the proofs of [19] become particularly simple in dimension 3.</p>