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May  2017, 22(3): 895-911. doi: 10.3934/dcdsb.2017045

## Asymptotic profile of the solution to a free boundary problem arising in a shifting climate model

 1 Department of Mathematics, University of Science and Technology of China, Hefei 230026, China 2 School of Science and Technology, University of New England, Armidale, NSW 2351, Australia

Received  September 2015 Revised  November 2015 Published  January 2017

Fund Project: C. Lei is supported by the China Scholarship Council for 2 years of study at University of New England, Y. Du was supported by the Australian Research Council.

We give a complete description of the long-time asymptotic profile of the solution to a free boundary model considered recently in [10]. This model describes the spreading of an invasive species in an environment which shifts with a constant speed, and the research of [10] 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 [10].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.

Citation: Chengxia Lei, Yihong Du. Asymptotic profile of the solution to a free boundary problem arising in a shifting climate model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 895-911. doi: 10.3934/dcdsb.2017045
##### References:
 [1] S. B. Angenent, The zero set of a solution of a parabolic equation, J. Reine Angew. Math., 390 (1988), 79-96.  doi: 10.1515/crll.1988.390.79.  Google Scholar [2] H. Berestycki, O. Diekmann, C. J. Nagelkerke and P. A. Zegeling, Can a species keep pace with a shifting climate?, Bull. Math. Biol., 71 (2009), 399-429.  doi: 10.1007/s11538-008-9367-5.  Google Scholar [3] J. Cai, B. Lou and M. Zhou, Asymptotic behavior of solutions of a reaction diffusion equation with free boundary conditions, J. Dynam. Differential Equations, 26 (2014), 1007-1028.  doi: 10.1007/s10884-014-9404-z.  Google Scholar [4] E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations McGraw-Hill, New York, 1955. Google Scholar [5] Y. Du and Z. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377-405.  doi: 10.1137/090771089.  Google Scholar [6] Y. Du and B. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc. (JEMS), 17 (2015), 2673-2724.  doi: 10.4171/JEMS/568.  Google Scholar [7] Y. Du, B. Lou and M. Zhou, Nonlinear diffusion problems with free boundaries: Convergence, transition speed and zero number arguments, SIAM J. Math. Anal., 47 (2015), 3555-3584.  doi: 10.1137/140994848.  Google Scholar [8] Y. Du and L. Ma, Logistic type equations on $\mathbb{R}^N$ by a squeezing method involving boundary blow-up solutions, J. London Math. Soc., 64 (2001), 107-124.  doi: 10.1017/S0024610701002289.  Google Scholar [9] Y. Du, H. Matsuzawa and M. Zhou, Sharp estimate of the spreading speed determined by nonlinear free boundary problems, SIAM J. Math. Anal., 46 (2014), 375-396.  doi: 10.1137/130908063.  Google Scholar [10] Y. Du, L. Wei and L. Zhou, Spreading in a shifting environment modeled by the diffusive logistic equation with a free boundary, Preprint, arXiv: 1508.06246 Google Scholar [11] Y. Du, M. Wang and M. Zhou, Semi-wave and spreading speed for the diffusive competition model with a free boundary J. Math. Pures Appl. , in press. Google Scholar [12] H. Gu, B. Lou and M. Zhou, Long time behavior of solutions of Fisher-KPP equation with advection and free boundaries, J. Funct. Anal., 269 (2015), 1714-1768.  doi: 10.1016/j.jfa.2015.07.002.  Google Scholar [13] Y. Kaneko and H. Matsuzawa, Spreading speed and sharp asymptotic profiles of solutions in free boundary problems for nonlinear advection-diffusion equations, J. Math. Anal. Appl., 428 (2015), 43-76.  doi: 10.1016/j.jmaa.2015.02.051.  Google Scholar [14] B. Li, S. Bewick, J. Shang and W. Fagan, Persistence and spread of a species with a shifting habitat edge, SIAM J. Appl. Math., 74 (2014), 1397-1417.  doi: 10.1137/130938463.  Google Scholar

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##### References:
 [1] S. B. Angenent, The zero set of a solution of a parabolic equation, J. Reine Angew. Math., 390 (1988), 79-96.  doi: 10.1515/crll.1988.390.79.  Google Scholar [2] H. Berestycki, O. Diekmann, C. J. Nagelkerke and P. A. Zegeling, Can a species keep pace with a shifting climate?, Bull. Math. Biol., 71 (2009), 399-429.  doi: 10.1007/s11538-008-9367-5.  Google Scholar [3] J. Cai, B. Lou and M. Zhou, Asymptotic behavior of solutions of a reaction diffusion equation with free boundary conditions, J. Dynam. Differential Equations, 26 (2014), 1007-1028.  doi: 10.1007/s10884-014-9404-z.  Google Scholar [4] E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations McGraw-Hill, New York, 1955. Google Scholar [5] Y. Du and Z. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377-405.  doi: 10.1137/090771089.  Google Scholar [6] Y. Du and B. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc. (JEMS), 17 (2015), 2673-2724.  doi: 10.4171/JEMS/568.  Google Scholar [7] Y. Du, B. Lou and M. Zhou, Nonlinear diffusion problems with free boundaries: Convergence, transition speed and zero number arguments, SIAM J. Math. Anal., 47 (2015), 3555-3584.  doi: 10.1137/140994848.  Google Scholar [8] Y. Du and L. Ma, Logistic type equations on $\mathbb{R}^N$ by a squeezing method involving boundary blow-up solutions, J. London Math. Soc., 64 (2001), 107-124.  doi: 10.1017/S0024610701002289.  Google Scholar [9] Y. Du, H. Matsuzawa and M. Zhou, Sharp estimate of the spreading speed determined by nonlinear free boundary problems, SIAM J. Math. Anal., 46 (2014), 375-396.  doi: 10.1137/130908063.  Google Scholar [10] Y. Du, L. Wei and L. Zhou, Spreading in a shifting environment modeled by the diffusive logistic equation with a free boundary, Preprint, arXiv: 1508.06246 Google Scholar [11] Y. Du, M. Wang and M. Zhou, Semi-wave and spreading speed for the diffusive competition model with a free boundary J. Math. Pures Appl. , in press. Google Scholar [12] H. Gu, B. Lou and M. Zhou, Long time behavior of solutions of Fisher-KPP equation with advection and free boundaries, J. Funct. Anal., 269 (2015), 1714-1768.  doi: 10.1016/j.jfa.2015.07.002.  Google Scholar [13] Y. Kaneko and H. Matsuzawa, Spreading speed and sharp asymptotic profiles of solutions in free boundary problems for nonlinear advection-diffusion equations, J. Math. Anal. Appl., 428 (2015), 43-76.  doi: 10.1016/j.jmaa.2015.02.051.  Google Scholar [14] B. Li, S. Bewick, J. Shang and W. Fagan, Persistence and spread of a species with a shifting habitat edge, SIAM J. Appl. Math., 74 (2014), 1397-1417.  doi: 10.1137/130938463.  Google Scholar
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