July  2015, 35(7): 3253-3276. doi: 10.3934/dcds.2015.35.3253

Asymptotic behavior of solutions for competitive models with a free boundary

1. 

Department of Mathematics, Tongji University, Shanghai, 200092

Received  June 2014 Revised  November 2014 Published  January 2015

In this paper, we study a competitive model involving two species separated by a free boundary by virtue of strong competition. When the initial data has positive lower bounds near $\pm\infty$, we prove that the solution converges, as $t\rightarrow \infty$, to a traveling wave solution and the free boundary moves to infinity with a constant speed.
Citation: Jian Yang. Asymptotic behavior of solutions for competitive models with a free boundary. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3253-3276. doi: 10.3934/dcds.2015.35.3253
References:
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J. J. Cai, B. D. Lou and M. L. Zhou, Asymptotic behavior of solutions of a reaction diffusion equation with free boundary conditions,, J. Dynam. Differential Equations, 26 (2014), 1007.  doi: 10.1007/s10884-014-9404-z.  Google Scholar

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C. H. Chang and C. C. Chen, Travelling wave solutions of a free boundary problem for a two-species competitive model,, Commun. Pure Appl. Anal., 12 (2013), 1065.  doi: 10.3934/cpaa.2013.12.1065.  Google Scholar

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X. F. Chen, B. D. Lou, M. L. Zhou and T. Giletti, Long time behavior of solutions of a reaction-diffusion equation on unbounded intervals with Robin boundary conditions,, Ann. Inst. H. Poincaré Anal. Non Linéaire, (2014).  doi: 10.1016/j.anihpc.2014.08.004.  Google Scholar

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Y. H. Du, H. Matsuzawa and M. L. Zhou, Spreading speed determined by nonlinear free boundary problems in high dimensions,, J. Math. Pures Appl., ().   Google Scholar

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P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions,, Arch. Ration. Mech. Anal., 65 (1977), 335.   Google Scholar

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O. A. Ladyzenskaya, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type,, Academic Press, (1968).   Google Scholar

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G. M. Lieberman, Second Order Parabolic Differential Equations,, World Scientific, (1996).  doi: 10.1142/3302.  Google Scholar

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M. Mimura, Y. Yamada and S. Yotsutani, A free boundary problem in ecology,, Japan J. Appl. Math., 2 (1985), 151.  doi: 10.1007/BF03167042.  Google Scholar

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M. Mimura, Y. Yamada and S. Yotsutani, Stability analysis for free boundary problems in ecology,, Hiroshima Math. J., 16 (1986), 477.   Google Scholar

[12]

M. Mimura, Y. Yamada and S. Yotsutani, Free boundary problems for some reaction-diffusion equations,, Hiroshima Math. J., 17 (1987), 241.   Google Scholar

[13]

J. Yang and B. D. Lou, Traveling wave solutions of competitive models with free boundaries,, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 817.  doi: 10.3934/dcdsb.2014.19.817.  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.  doi: 10.1515/crll.1988.390.79.  Google Scholar

[2]

J. J. Cai, B. D. Lou and M. L. Zhou, Asymptotic behavior of solutions of a reaction diffusion equation with free boundary conditions,, J. Dynam. Differential Equations, 26 (2014), 1007.  doi: 10.1007/s10884-014-9404-z.  Google Scholar

[3]

C. H. Chang and C. C. Chen, Travelling wave solutions of a free boundary problem for a two-species competitive model,, Commun. Pure Appl. Anal., 12 (2013), 1065.  doi: 10.3934/cpaa.2013.12.1065.  Google Scholar

[4]

X. F. Chen, B. D. Lou, M. L. Zhou and T. Giletti, Long time behavior of solutions of a reaction-diffusion equation on unbounded intervals with Robin boundary conditions,, Ann. Inst. H. Poincaré Anal. Non Linéaire, (2014).  doi: 10.1016/j.anihpc.2014.08.004.  Google Scholar

[5]

Y. H. Du and Z. G. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary,, SIAM J. Math. Anal., 42 (2010), 377.  doi: 10.1137/090771089.  Google Scholar

[6]

Y. H. Du, H. Matsuzawa and M. L. Zhou, Spreading speed determined by nonlinear free boundary problems in high dimensions,, J. Math. Pures Appl., ().   Google Scholar

[7]

P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions,, Arch. Ration. Mech. Anal., 65 (1977), 335.   Google Scholar

[8]

O. A. Ladyzenskaya, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type,, Academic Press, (1968).   Google Scholar

[9]

G. M. Lieberman, Second Order Parabolic Differential Equations,, World Scientific, (1996).  doi: 10.1142/3302.  Google Scholar

[10]

M. Mimura, Y. Yamada and S. Yotsutani, A free boundary problem in ecology,, Japan J. Appl. Math., 2 (1985), 151.  doi: 10.1007/BF03167042.  Google Scholar

[11]

M. Mimura, Y. Yamada and S. Yotsutani, Stability analysis for free boundary problems in ecology,, Hiroshima Math. J., 16 (1986), 477.   Google Scholar

[12]

M. Mimura, Y. Yamada and S. Yotsutani, Free boundary problems for some reaction-diffusion equations,, Hiroshima Math. J., 17 (1987), 241.   Google Scholar

[13]

J. Yang and B. D. Lou, Traveling wave solutions of competitive models with free boundaries,, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 817.  doi: 10.3934/dcdsb.2014.19.817.  Google Scholar

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