Spreading in space-time periodic media governed by a monostable equation with free boundaries, Part 1: Continuous initial functions

TL;DR

This paper studies the long-term behavior of solutions to a free boundary problem modeling species spread in space-time periodic environments, extending classical theory to continuous initial data and establishing a spreading-vanishing dichotomy.

Contribution

It extends the existence and uniqueness theory of solutions to include continuous initial data and develops methods to analyze spreading speed without semi-wave solutions.

Findings

Established existence and uniqueness of classical solutions for continuous initial data.

Proved a spreading-vanishing dichotomy for the model.

Extended the theoretical framework beyond $C^2$ initial data.

Abstract

We aim to classify the long-time behavior of the solution to a free boundary problem with monostable reaction term in space-time periodic media. Such a model may be used to describe the spreading of a new or invasive species, with the free boundary representing the expanding front. In time-periodic and space homogeneous environment, as well as in space-periodic and time autonomous environment, such a problem has been studied recently in \cite{dgp, dl}. In both cases, a spreading-vanishing dichotomy has been established, and when spreading happens, the asymptotic spreading speed is proved to exist by making use of the corresponding semi-wave solutions. The approaches in \cite{dgp, dl} seem difficult to apply to the current situation where the environment is periodic in both space and time. Here we take a different approach, based on the methods developed by Weinberger \cite{w1, w2} and others \cite{fyz,lyz,lz1,lz2,lui}, which yield the existence of the spreading speed without using traveling wave solutions. In Part 1 of this work, we establish the existence and uniqueness of classical solutions for the free boundary problem with continuous initial data, extending the existing theory which was established only for $C^2$ initial data. This will enable us to develop Weinberger's method in Part 2 to determine the spreading speed without knowing a priori the existence of the corresponding semi-wave solutions. In Part 1 here, we also establish a spreading-vanishing dichotomy.