The high dimensional monostable reaction-diffusion equation with free boundary and radial symmetry
Abstract
We consider the radially symmetric version of the reaction-diffusion equation ut−dΔu=f(u) with a monostable nonlinearity f, viewed as a model for the spreading of a species with population range r<h(t) and density u(t,r) (r=|x|), where the free boundary r=h(t) is governed by u(t,h(t))=δ>0 and h'(t)=−d ur(t,h(t))/δ. For the one-dimensional case (N=1), Du \cite{DN} proved that when δ∈(0,1), spreading occurs: u→1 locally uniformly in ℝ, h(t)→∞, and limt→∞[h(t)−c*t]=h̃∈ℝ with no logarithmic shift. In the present paper we consider N≥2 and establish a complete trichotomy: spreading for δ∈(0,1); transition for δ=1, where u→1 uniformly on [0,h(t)] and h(t)→ h∞∈(0,∞); and vanishing for δ>1, where h(t)→0 and u→δ uniformly on [0,h(t)]. For the spreading regime, by constructing sharp upper and lower solutions, we prove that the solution converges globally to the semi-wave profile and reveal a logarithmic shift of the form $ \lim_{t\to\infty}\big[h(t)-c_*t+c_N(δ)\log t\big]=\hat{h}\in\mathbb{R}$, with the coefficient cN(δ)>0 satisfying $ \lim_{δ\to0}c_N(δ)=d(N-1)/c_0$, where d(N−1)/c0 is the shift coefficient for the high-dimensional radial pushed-case Cauchy problem. These results reveal the connection to the spreading behavior modeled by the corresponding Cauchy problem.
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BibTeX
@misc{cao2026high,
title = {{The high dimensional monostable reaction-diffusion equation with free boundary and radial symmetry}},
author = {Hongkai Cao and Jingyi Cui and Chengzhe Tang and Xiaoyan Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.25672},
eprint = {2608.25672},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.25672},
}
RIS
TY - UNPB
TI - The high dimensional monostable reaction-diffusion equation with free boundary and radial symmetry
AU - Hongkai Cao
AU - Jingyi Cui
AU - Chengzhe Tang
AU - Xiaoyan Zhang
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2608.25672
UR - https://arxiv.org/abs/2608.25672
AB - We consider the radially symmetric version of the reaction-diffusion equation u_t−dΔu=f(u) with a monostable nonlinearity f, viewed as a model for the spreading of a species with population range r<h(t) and density u(t,r) (r=|x|), where the free boundary r=h(t) is governed by u(t,h(t))=δ>0 and h'(t)=−d u_r(t,h(t))/δ. For the one-dimensional case (N=1), Du \cite{DN} proved that when δ∈(0,1), spreading occurs: u→1 locally uniformly in ℝ, h(t)→∞, and lim_t→∞[h(t)−c_*t]=h̃∈ℝ with no logarithmic shift. In the present paper we consider N≥2 and establish a complete trichotomy: spreading for δ∈(0,1); transition for δ=1, where u→1 uniformly on [0,h(t)] and h(t)→ h_∞∈(0,∞); and vanishing for δ>1, where h(t)→0 and u→δ uniformly on [0,h(t)]. For the spreading regime, by constructing sharp upper and lower solutions, we prove that the solution converges globally to the semi-wave profile and reveal a logarithmic shift of the form $ \lim_{t\to\infty}\big[h(t)-c_*t+c_N(δ)\log t\big]=\hat{h}\in\mathbb{R}$, with the coefficient c_N(δ)>0 satisfying $ \lim_{δ\to0}c_N(δ)=d(N-1)/c_0$, where d(N−1)/c_0 is the shift coefficient for the high-dimensional radial pushed-case Cauchy problem. These results reveal the connection to the spreading behavior modeled by the corresponding Cauchy problem.
ER -