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Isolated singularity for semilinear elliptic equations
Asymptotic behavior of solutions for competitive models with a free boundary
1.  Department of Mathematics, Tongji University, Shanghai, 200092 
References:
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S. B. Angenent, The zero set of a solution of a parabolic equation, J. Reine Angew. Math., 390 (1988), 7996. 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), 10071028. doi: 10.1007/s108840149404z. 
[3] 
C. H. Chang and C. C. Chen, Travelling wave solutions of a free boundary problem for a twospecies competitive model, Commun. Pure Appl. Anal., 12 (2013), 10651074. 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 reactiondiffusion 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, Spreadingvanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377405. 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), 335361. 
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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. 
[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), 151186. 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), 477498. 
[12] 
M. Mimura, Y. Yamada and S. Yotsutani, Free boundary problems for some reactiondiffusion equations, Hiroshima Math. J., 17 (1987), 241280. 
[13] 
J. Yang and B. D. Lou, Traveling wave solutions of competitive models with free boundaries, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 817826. 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), 7996. 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), 10071028. doi: 10.1007/s108840149404z. 
[3] 
C. H. Chang and C. C. Chen, Travelling wave solutions of a free boundary problem for a twospecies competitive model, Commun. Pure Appl. Anal., 12 (2013), 10651074. 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 reactiondiffusion 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, Spreadingvanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377405. 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), 335361. 
[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), 151186. 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), 477498. 
[12] 
M. Mimura, Y. Yamada and S. Yotsutani, Free boundary problems for some reactiondiffusion equations, Hiroshima Math. J., 17 (1987), 241280. 
[13] 
J. Yang and B. D. Lou, Traveling wave solutions of competitive models with free boundaries, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 817826. doi: 10.3934/dcdsb.2014.19.817. 
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