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Global Carleman estimate for the Kawahara equation and its applications
A free boundary problem for the FisherKPP equation with a given moving boundary
National Institute of Technology, Numazu College, 3600 Ooka, Numazu City, Shizuoka 4108501, Japan 
We study free boundary problem of FisherKPP equation $u_t = u_{xx}+u(1u), \ t>0, \ ct<x<h(t)$. The number $c>0$ is a given constant, $h(t)$ is a free boundary which is determined by the Stefanlike condition. This model may be used to describe the spreading of a nonnative 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 FisherKPP equation with a shiftingenvironment, which was considered by Du, Wei and Zhou [
References:
[1] 
S. B. Angenent, The zero set of a solution of a parabolic equation, J. Reine Angew. Math., 390 (1988), 7996. Google Scholar 
[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), 549. Google Scholar 
[3] 
D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 3376. Google Scholar 
[4] 
J. Cai, Asymptotic behavior of solutions of FisherKPP equation with free boundary conditions, Nonlinear Anal., 16 (2014), 170177. Google Scholar 
[5] 
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), 10071028. Google Scholar 
[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. Google Scholar 
[7] 
Y. Du and Z. Lin, Spreadingvanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377405. Google Scholar 
[8] 
Y. Du and B. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc., 17 (2015), 26732724. Google Scholar 
[9] 
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), 35553584. Google Scholar 
[10] 
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), 375396. Google Scholar 
[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/s1088401796142. Google Scholar 
[12] 
F. J. Fernandez, Unique continuation for parabolic operators. Ⅱ, Comm. Partial Differential Equations, 28 (2003), 15971604. Google Scholar 
[13] 
H. Gu, B. Lou and M. Zhou, Long time behavior of solutions of FisherKPP equation with advection and free boundaries, J. Funct. Anal., 269 (2015), 17141768. Google Scholar 
[14] 
Y. Kaneko, K. Oeda and Y. Yamada, Remarks on spreading and vanishing for free boundary problems of some reactiondiffusion equations, Funkcial. Ekvac., 57 (2014), 449465. Google Scholar 
[15] 
Y. Kaneko and Y. Yamada, A free boundary problem for a reactiondiffusion equation appearing in ecology, Adv. Math. Sci. Appl., 21 (2011), 467492. Google Scholar 
[16] 
Y. Kaneko and H. Matsuzawa, Spreading speed and sharp asymptotic profiles of solutions in free boundary problems for nonlinear advectiondiffusion equations, J. Math. Anal. Appl., 428 (2015), 4376. Google Scholar 
[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. Google Scholar 
[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. Google Scholar 
[19] 
G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific, Singapore, 1996. 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), 7996. Google Scholar 
[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), 549. Google Scholar 
[3] 
D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 3376. Google Scholar 
[4] 
J. Cai, Asymptotic behavior of solutions of FisherKPP equation with free boundary conditions, Nonlinear Anal., 16 (2014), 170177. Google Scholar 
[5] 
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), 10071028. Google Scholar 
[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. Google Scholar 
[7] 
Y. Du and Z. Lin, Spreadingvanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377405. Google Scholar 
[8] 
Y. Du and B. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc., 17 (2015), 26732724. Google Scholar 
[9] 
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), 35553584. Google Scholar 
[10] 
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), 375396. Google Scholar 
[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/s1088401796142. Google Scholar 
[12] 
F. J. Fernandez, Unique continuation for parabolic operators. Ⅱ, Comm. Partial Differential Equations, 28 (2003), 15971604. Google Scholar 
[13] 
H. Gu, B. Lou and M. Zhou, Long time behavior of solutions of FisherKPP equation with advection and free boundaries, J. Funct. Anal., 269 (2015), 17141768. Google Scholar 
[14] 
Y. Kaneko, K. Oeda and Y. Yamada, Remarks on spreading and vanishing for free boundary problems of some reactiondiffusion equations, Funkcial. Ekvac., 57 (2014), 449465. Google Scholar 
[15] 
Y. Kaneko and Y. Yamada, A free boundary problem for a reactiondiffusion equation appearing in ecology, Adv. Math. Sci. Appl., 21 (2011), 467492. Google Scholar 
[16] 
Y. Kaneko and H. Matsuzawa, Spreading speed and sharp asymptotic profiles of solutions in free boundary problems for nonlinear advectiondiffusion equations, J. Math. Anal. Appl., 428 (2015), 4376. Google Scholar 
[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. Google Scholar 
[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. Google Scholar 
[19] 
G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific, Singapore, 1996. Google Scholar 
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