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September 2018, 17(5): 1821-1852. doi: 10.3934/cpaa.2018087

A free boundary problem for the Fisher-KPP equation with a given moving boundary

National Institute of Technology, Numazu College, 3600 Ooka, Numazu City, Shizuoka 410-8501, Japan

Received  July 2017 Revised  December 2017 Published  April 2018

Fund Project: The author was partly supported by JSPS KAKENHI Grant-in-Aid for Scientific Research (C) 17K05340

We study free boundary problem of Fisher-KPP equation $u_t = u_{xx}+u(1-u), \ t>0, \ ct<x<h(t)$. The number $c>0$ is a given constant, $h(t)$ is a free boundary which is determined by the Stefan-like condition. This model may be used to describe the spreading of a non-native species over a one dimensional habitat. The free boundary $x = h(t)$ represents the spreading front. In this model, we impose zero Dirichlet condition at left moving boundary $x = ct$. This means that the left boundary of the habitat is a very hostile environment and that the habitat is eroded away by the left moving boundary at constant speed $c$.

In this paper we will give a trichotomy result, that is, for any initial data, exactly one of the three behaviors, vanishing, spreading and transition, happens. This result is related to the results appear in the free boundary problem for the Fisher-KPP equation with a shifting-environment, which was considered by Du, Wei and Zhou [11]. However the vanishing in our problem is different from that in [11] because in our vanishing case, the solution is not global-in-time.

Citation: Hiroshi Matsuzawa. A free boundary problem for the Fisher-KPP equation with a given moving boundary. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1821-1852. doi: 10.3934/cpaa.2018087
References:
[1]

S. B. Angenent, The zero set of a solution of a parabolic equation, J. Reine Angew. Math., 390 (1988), 79-96.

[2]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in Partial Differential Equations and Related Topics, Lecture Notes in Mathematics. 446, Springer, Berlin, (1975), 5-49.

[3]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76.

[4]

J. Cai, Asymptotic behavior of solutions of Fisher-KPP equation with free boundary conditions, Nonlinear Anal., 16 (2014), 170-177.

[5]

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.

[6]

Y. Du, Order Structure and Topological Methods in Nonlinear Partial Differential Equations, Vol. 1 Maximum Principle and Applications, World Scientific Publishing, 2006. doi: 10.1142/5999.

[7]

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.

[8]

Y. Du and B. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc., 17 (2015), 2673-2724.

[9]

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.

[10]

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.

[11]

Y. Du, L. Wei and L. Zhou, Spreading in a shifting environment modeled by the diffusive logistic equation with a free boundary, J. Dynam. Differential Equations, (2017). doi: 10.1007/s10884-017-9614-2.

[12]

F. J. Fernandez, Unique continuation for parabolic operators. Ⅱ, Comm. Partial Differential Equations, 28 (2003), 1597-1604.

[13]

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.

[14]

Y. KanekoK. Oeda and Y. Yamada, Remarks on spreading and vanishing for free boundary problems of some reaction-diffusion equations, Funkcial. Ekvac., 57 (2014), 449-465.

[15]

Y. Kaneko and Y. Yamada, A free boundary problem for a reaction-diffusion equation appearing in ecology, Adv. Math. Sci. Appl., 21 (2011), 467-492.

[16]

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.

[17]

Y. Kaneko and H. Matsuzawa, Spreading and vanishing in a free boundary problem for nonlinear diffusion equations with a given forced moving boundary, J. Differential Equations, to appear. doi: 10.1016/j.jde.2018.03.026.

[18]

O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural' ceva, Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc., Providence, RI, 1968.

[19]

G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific, Singapore, 1996.

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.

[2]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in Partial Differential Equations and Related Topics, Lecture Notes in Mathematics. 446, Springer, Berlin, (1975), 5-49.

[3]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76.

[4]

J. Cai, Asymptotic behavior of solutions of Fisher-KPP equation with free boundary conditions, Nonlinear Anal., 16 (2014), 170-177.

[5]

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.

[6]

Y. Du, Order Structure and Topological Methods in Nonlinear Partial Differential Equations, Vol. 1 Maximum Principle and Applications, World Scientific Publishing, 2006. doi: 10.1142/5999.

[7]

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.

[8]

Y. Du and B. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc., 17 (2015), 2673-2724.

[9]

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.

[10]

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.

[11]

Y. Du, L. Wei and L. Zhou, Spreading in a shifting environment modeled by the diffusive logistic equation with a free boundary, J. Dynam. Differential Equations, (2017). doi: 10.1007/s10884-017-9614-2.

[12]

F. J. Fernandez, Unique continuation for parabolic operators. Ⅱ, Comm. Partial Differential Equations, 28 (2003), 1597-1604.

[13]

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.

[14]

Y. KanekoK. Oeda and Y. Yamada, Remarks on spreading and vanishing for free boundary problems of some reaction-diffusion equations, Funkcial. Ekvac., 57 (2014), 449-465.

[15]

Y. Kaneko and Y. Yamada, A free boundary problem for a reaction-diffusion equation appearing in ecology, Adv. Math. Sci. Appl., 21 (2011), 467-492.

[16]

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.

[17]

Y. Kaneko and H. Matsuzawa, Spreading and vanishing in a free boundary problem for nonlinear diffusion equations with a given forced moving boundary, J. Differential Equations, to appear. doi: 10.1016/j.jde.2018.03.026.

[18]

O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural' ceva, Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc., Providence, RI, 1968.

[19]

G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific, Singapore, 1996.

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