# American Institute of Mathematical Sciences

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:
 [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. [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. [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. [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. [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. [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., (). [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. [8] O. A. Ladyzenskaya, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type,, Academic Press, (1968). [9] G. M. Lieberman, Second Order Parabolic Differential Equations,, World Scientific, (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. 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. [12] M. Mimura, Y. Yamada and S. Yotsutani, Free boundary problems for some reaction-diffusion equations,, Hiroshima Math. J., 17 (1987), 241. [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.

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##### 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. [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. [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. [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. [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. [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., (). [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. [8] O. A. Ladyzenskaya, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type,, Academic Press, (1968). [9] G. M. Lieberman, Second Order Parabolic Differential Equations,, World Scientific, (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. 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. [12] M. Mimura, Y. Yamada and S. Yotsutani, Free boundary problems for some reaction-diffusion equations,, Hiroshima Math. J., 17 (1987), 241. [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.
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