##
**Asymptotic profile of the solution to a free boundary problem arising in a shifting climate model.**
*(English)*
Zbl 1360.35299

Summary: We give a complete description of the long-time asymptotic profile of the solution to a free boundary model considered recently in [the second author et al., “Spreading in a shifting environment modeled by the diffusive logistic equation with a free boundary”, Preprint, arXiv:1508.06246]. This model describes the spreading of an invasive species in an environment which shifts with a constant speed, and the research of [loc. cit.] indicates that the species may vanish, or spread successfully, or fall in a borderline case. In the case of successful spreading, the long-time behavior of the population is not completely understood in [loc. cit.]. Here we show that the spreading of the species is governed by two traveling waves, one has the speed of the shifting environment, giving the profile of the retreating tail of the population, while the other has a faster speed determined by a semi-wave, representing the profile of the advancing front of the population.

### MSC:

35Q92 | PDEs in connection with biology, chemistry and other natural sciences |

35K20 | Initial-boundary value problems for second-order parabolic equations |

35R35 | Free boundary problems for PDEs |

92B05 | General biology and biomathematics |

PDF
BibTeX
XML
Cite

\textit{C. Lei} and \textit{Y. Du}, Discrete Contin. Dyn. Syst., Ser. B 22, No. 3, 895--911 (2017; Zbl 1360.35299)

Full Text:
DOI

### References:

[1] | S. B. Angenent, The zero set of a solution of a parabolic equation,, J. Reine Angew. Math., 390, 79, (1988) · Zbl 0644.35050 |

[2] | H. Berestycki, Can a species keep pace with a shifting climate?,, Bull. Math. Biol., 71, 399, (2009) · Zbl 1169.92043 |

[3] | J. Cai, Asymptotic behavior of solutions of a reaction diffusion equation with free boundary conditions,, J. Dynam. Differential Equations, 26, 1007, (2014) · Zbl 1328.35329 |

[4] | E. A. Coddington, <em>Theory of Ordinary Differential Equations</em>,, McGraw-Hill, (1955) · Zbl 0064.33002 |

[5] | Y. Du, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary,, SIAM J. Math. Anal., 42, 377, (2010) · Zbl 1219.35373 |

[6] | Y. Du, Spreading and vanishing in nonlinear diffusion problems with free boundaries,, J. Eur. Math. Soc. (JEMS), 17, 2673, (2015) · Zbl 1331.35399 |

[7] | Y. Du, Nonlinear diffusion problems with free boundaries: Convergence, transition speed and zero number arguments,, SIAM J. Math. Anal., 47, 3555, (2015) · Zbl 1321.35086 |

[8] | Y. Du, Logistic type equations on \(\mathbbR^N\) by a squeezing method involving boundary blow-up solutions,, J. London Math. Soc., 64, 107, (2001) · Zbl 1018.35045 |

[9] | Y. Du, Sharp estimate of the spreading speed determined by nonlinear free boundary problems,, SIAM J. Math. Anal., 46, 375, (2014) · Zbl 1296.35219 |

[10] | Y. Du, Spreading in a shifting environment modeled by the diffusive logistic equation with a free boundary, Preprint,, <a href= |

[11] | Y. Du, Semi-wave and spreading speed for the diffusive competition model with a free boundary,, J. Math. Pures Appl. · Zbl 1377.35136 |

[12] | H. Gu, Long time behavior of solutions of Fisher-KPP equation with advection and free boundaries,, J. Funct. Anal., 269, 1714, (2015) · Zbl 1335.35102 |

[13] | Y. Kaneko, Spreading speed and sharp asymptotic profiles of solutions in free boundary problems for nonlinear advection-diffusion equations,, J. Math. Anal. Appl., 428, 43, (2015) · Zbl 1325.35292 |

[14] | B. Li, Persistence and spread of a species with a shifting habitat edge,, SIAM J. Appl. Math., 74, 1397, (2014) · Zbl 1345.92120 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.