Precise propagation profile for some monostable free boundary problems in time-periodic media

TL;DR

This work analyzes a time-periodic, one-dimensional reaction-diffusion equation with free boundaries modeling population range expansion under a periodic environment. It proves the existence and uniqueness of a time-periodic semi-wave pair $$(k^*(t), \Phi^*(t,x))$$ for general monostable nonlinearities, and establishes sharp convergence of the free boundary solution to a translate of the semi-wave without relying on the KPP condition. The authors develop a robust framework based on upper and lower solutions, to obtain precise bounds on the moving boundaries $g(t)$ and $h(t)$ and to show that, as $t\to\infty$, $u(t,x)$ converges to $\Phi^*(t,h(t)-x)$ on the right and $\Phi^*(t,x-g(t))$ on the left, with the speed given by the mean $\overline{k^*}=rac{1}{T}\int_0^T k^*(s)\,ds$. By identifying the limiting pair $(V,\Gamma)$ with the semi-wave, the paper delivers a complete sharp description of the spreading dynamics in heterogeneous, time-periodic media, extending previous autonomous or strong-KPP results to general monostable nonlinearities and illustrating the broader applicability of semi-wave techniques in free boundary problems.

Abstract

We consider reaction-diffusion equations of the form \begin{equation*} u_t - d u_{xx} = f(t,u), \quad t>0,\ \ x \in [g(t), h(t)], \end{equation*} where $f(t,u)$ is periodic in $t$ and monostable in $u$, and the interval $[g(t), h(t)]$ represents the one dimensional population range of a species with density $u(t,x)$ at time $t$ and spatial location $x$. The free boundaries $x=g(t)$ and $x=h(t)$ evolve subject to a ``preferred population density" condition at the habitat edges. Analogous to the traveling wave solutions in the corresponding Cauchy problem, semi-wave solutions play a fundamental role in understanding the propagation phenomena governed by the free boundary problem here. But in contrast to the Cauchy problem, where the KPP condition plays a subtle role in the precise approximation of its solution (with compactly supported initial function) by the traveling wave solution with minimal speed, here we prove the existence and uniqueness of a semi-wave in a general monostable setting, and obtain a precise description of the convergence of the solution toward the semi-wave as time goes to infinity, where the KPP condition plays no special role. Previously, such a sharp result was proved for a free boundary model only when $f$ is autonomous ($f=f(u)$, see \cite{D} or \cite{DL15} for a related free boundary model), or a less precise result was obtained in the time-periodic case under an extra strong KPP condition on $f$ (see \cite{MDW}, or \cite{DGP} for a related free boundary model). This work appears to be the first to prove the sharp convergence result for a general monostable free boundary problem in a heterogeneous environment, and we believe the methods developed here should have applications to related free boundary problems in heterogeneous media with nonlinearities more general than those of KPP type.

Theorems & Definitions (22)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• proof
• Lemma 2.1
• Lemma 2.2
• Proposition 2.3
• proof : Proof of Theorem \ref{['theo-semi-wave']}
• Lemma 3.1
• proof