Diffusive KPP equations with free boundaries in time almost periodic environments: I. Spreading and vanishing dichotomy

Fang  Li; Xing  Liang; Wenxian Shen

doi:10.3934/dcds.2016.36.3317

Article Contents

2016, Volume 36, Issue 6: 3317-3338. Doi: 10.3934/dcds.2016.36.3317

This issue Previous Article Sharp criteria of Liouville type for some nonlinear systems Next Article On two-sided estimates for the nonlinear Fourier transform of KdV

Diffusive KPP equations with free boundaries in time almost periodic environments: I. Spreading and vanishing dichotomy

1.
School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui, 230026
2.
Department of Mathematics and Statistics, Auburn University, Auburn University, AL 36849-5310

Received: May 2015

Revised: October 2015

Published: May 2017

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Abstract

In this series of papers, we investigate the spreading and vanishing dynamics of time almost periodic diffusive KPP equations with free boundaries. Such equations are used to characterize the spreading of a new species in time almost periodic environments with free boundaries representing the spreading fronts. In this first part, we show that a spreading-vanishing dichotomy occurs for such free boundary problems, that is, the species either successfully spreads to all the new environment and stabilizes at a time almost periodic positive solution, or it fails to establish and dies out eventually. The results of this part extend the existing results on spreading-vanishing dichotomy for time and space independent, or time periodic and space independent, or time independent and space periodic diffusive KPP equations with free boundaries. The extension is nontrivial and is ever done for the first time.

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Mathematics Subject Classification: 35K20, 35R35, 35K57, 35B15, 92B05.

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