• Previous Article
    Transboundary capital and pollution flows and the emergence of regional inequalities
  • DCDS-B Home
  • This Issue
  • Next Article
    Global existence and regularity results for strongly coupled nonregular parabolic systems via iterative methods
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. BerestyckiO. DiekmannC. 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. CaiB. 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. DuB. 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. DuH. 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. GuB. 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. LiS. BewickJ. 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

show all references

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. BerestyckiO. DiekmannC. 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. CaiB. 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. DuB. 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. DuH. 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. GuB. 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. LiS. BewickJ. 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

[1]

Harunori Monobe, Hirokazu Ninomiya. Traveling wave solutions with convex domains for a free boundary problem. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 905-914. doi: 10.3934/dcds.2017037

[2]

Jian Yang, Bendong Lou. Traveling wave solutions of competitive models with free boundaries. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 817-826. doi: 10.3934/dcdsb.2014.19.817

[3]

Harunori Monobe, Hirokazu Ninomiya. Multiple existence of traveling waves of a free boundary problem describing cell motility. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 789-799. doi: 10.3934/dcdsb.2014.19.789

[4]

Jesús Ildefonso Díaz, L. Tello. On a climate model with a dynamic nonlinear diffusive boundary condition. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 253-262. doi: 10.3934/dcdss.2008.1.253

[5]

Chang-Hong Wu. Spreading speed and traveling waves for a two-species weak competition system with free boundary. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2441-2455. doi: 10.3934/dcdsb.2013.18.2441

[6]

Feimin Huang, Xiaoding Shi, Yi Wang. Stability of viscous shock wave for compressible Navier-Stokes equations with free boundary. Kinetic & Related Models, 2010, 3 (3) : 409-425. doi: 10.3934/krm.2010.3.409

[7]

Chueh-Hsin Chang, Chiun-Chuan Chen. Travelling wave solutions of a free boundary problem for a two-species competitive model. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1065-1074. doi: 10.3934/cpaa.2013.12.1065

[8]

Guo Lin, Wan-Tong Li. Traveling wave solutions of a competitive recursion. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 173-189. doi: 10.3934/dcdsb.2012.17.173

[9]

Hayk Mikayelyan, Henrik Shahgholian. Convexity of the free boundary for an exterior free boundary problem involving the perimeter. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1431-1443. doi: 10.3934/cpaa.2013.12.1431

[10]

Toyohiko Aiki. A free boundary problem for an elastic material. Conference Publications, 2007, 2007 (Special) : 10-17. doi: 10.3934/proc.2007.2007.10

[11]

Avner Friedman. Free boundary problems arising in biology. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 193-202. doi: 10.3934/dcdsb.2018013

[12]

Fathi Dkhil, Angela Stevens. Traveling wave speeds in rapidly oscillating media. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 89-108. doi: 10.3934/dcds.2009.25.89

[13]

Bingtuan Li. Some remarks on traveling wave solutions in competition models. Discrete & Continuous Dynamical Systems - B, 2009, 12 (2) : 389-399. doi: 10.3934/dcdsb.2009.12.389

[14]

Wei Ding, Wenzhang Huang, Siroj Kansakar. Traveling wave solutions for a diffusive sis epidemic model. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1291-1304. doi: 10.3934/dcdsb.2013.18.1291

[15]

Vishal Vasan, Katie Oliveras. Pressure beneath a traveling wave with constant vorticity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3219-3239. doi: 10.3934/dcds.2014.34.3219

[16]

Jong-Shenq Guo, Ken-Ichi Nakamura, Toshiko Ogiwara, Chang-Hong Wu. The sign of traveling wave speed in bistable dynamics. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 0-0. doi: 10.3934/dcds.2020047

[17]

Xinfu Chen, Huibin Cheng. Regularity of the free boundary for the American put option. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1751-1759. doi: 10.3934/dcdsb.2012.17.1751

[18]

Xiaoshan Chen, Fahuai Yi. Free boundary problem of Barenblatt equation in stochastic control. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1421-1434. doi: 10.3934/dcdsb.2016003

[19]

Jia-Feng Cao, Wan-Tong Li, Fei-Ying Yang. Dynamics of a nonlocal SIS epidemic model with free boundary. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 247-266. doi: 10.3934/dcdsb.2017013

[20]

Avner Friedman. Free boundary problems for systems of Stokes equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1455-1468. doi: 10.3934/dcdsb.2016006

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (47)
  • HTML views (10)
  • Cited by (5)

Other articles
by authors

[Back to Top]