The Fisher-KPP nonlocal diffusion equation with free boundary and radial symmetry in \(\mathbb{R}^3 \). (English) Zbl 1535.35218
Summary: 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 [SIAM J. Math. Anal. 54, No. 3, 3930–3973 (2022; Zbl 1494.35033)], 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 [“The high dimensional Fisher-KPP nonlocal diffusion equation with free boundary and radial symmetry: sharp estimates”, Preprint, https://turing.une.edu.au/~ydu/papers/dn-highD-2-March2022.pdf], 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 [loc. cit.] become particularly simple in dimension 3.
Citations:
Zbl 1494.35033References:
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