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

Asymptotic behavior of solutions for competitive models with a free boundary

Jian Yang 1,

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.
Keywords: free boundary problem, competitive model, Asymptotic behavior, renormalization., reaction-diffusion equation.
Mathematics Subject Classification: Primary: 35B40, 35K57; Secondary: 35R3.
Citation: Jian Yang. Asymptotic behavior of solutions for competitive models with a free boundary. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 3253-3276. doi: 10.3934/dcds.2015.35.3253
References:
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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.

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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), 1007-1028. doi: 10.1007/s10884-014-9404-z.

[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), 1065-1074. doi: 10.3934/cpaa.2013.12.1065.

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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, in press, 2014. doi: 10.1016/j.anihpc.2014.08.004.

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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), 377-405. doi: 10.1137/090771089.

[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., preprint.

[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), 335-361.

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O. A. Ladyzenskaya, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Academic Press, New York, London, 1968.

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G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific, Singapore, 1996. doi: 10.1142/3302.

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M. Mimura, Y. Yamada and S. Yotsutani, A free boundary problem in ecology, Japan J. Appl. Math., 2 (1985), 151-186. doi: 10.1007/BF03167042.

[11]

M. Mimura, Y. Yamada and S. Yotsutani, Stability analysis for free boundary problems in ecology, Hiroshima Math. J., 16 (1986), 477-498.

[12]

M. Mimura, Y. Yamada and S. Yotsutani, Free boundary problems for some reaction-diffusion equations, Hiroshima Math. J., 17 (1987), 241-280.

[13]

J. Yang and B. D. Lou, Traveling wave solutions of competitive models with free boundaries, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 817-826. doi: 10.3934/dcdsb.2014.19.817.

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.

[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), 1007-1028. doi: 10.1007/s10884-014-9404-z.

[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), 1065-1074. doi: 10.3934/cpaa.2013.12.1065.

[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, in press, 2014. doi: 10.1016/j.anihpc.2014.08.004.

[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), 377-405. doi: 10.1137/090771089.

[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., preprint.

[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), 335-361.

[8]

O. A. Ladyzenskaya, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Academic Press, New York, London, 1968.

[9]

G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific, Singapore, 1996. doi: 10.1142/3302.

[10]

M. Mimura, Y. Yamada and S. Yotsutani, A free boundary problem in ecology, Japan J. Appl. Math., 2 (1985), 151-186. doi: 10.1007/BF03167042.

[11]

M. Mimura, Y. Yamada and S. Yotsutani, Stability analysis for free boundary problems in ecology, Hiroshima Math. J., 16 (1986), 477-498.

[12]

M. Mimura, Y. Yamada and S. Yotsutani, Free boundary problems for some reaction-diffusion equations, Hiroshima Math. J., 17 (1987), 241-280.

[13]

J. Yang and B. D. Lou, Traveling wave solutions of competitive models with free boundaries, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 817-826. doi: 10.3934/dcdsb.2014.19.817.

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