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

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:262

Article

Ding, Weiwei¹  Du, Yihong¹  Liang, Xing² 

[1] Univ New England, Sch Sci & Technol, Armidale, NSW 2351, Australia

[2] Univ Sci & Technol China, Sch Math Sci, Hefei 230026, Anhui, Peoples R China

关键词: Free boundary;    Space-time periodic media;    Spreading vanishing dichotomy;   

DOI  :  10.1016/j.jde.2017.01.016

来源: Elsevier

【 摘 要 】

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 [11,12]. 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 [11,12] 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 [31,32] and others [16,22-24,26], 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. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

Free