## 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

### Keywords:

shifting climate; free boundary; traveling wave
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### References:

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