AM  Vol.6 No.12 , November 2015
Spatial Segregation Limit of a Quasilinear Competition-Diffusion System
ABSTRACT
The aim of this paper is to investigate a Volterra-Lotka competition model of quasilinear parabolic equations with large interaction. Some existence, uniqueness and convergence results for the system are given. Also investigated is its spatial segregation limit when the interspecific competition rates become large. We show that the limit problem is similar to a free boundary problem.

Cite this paper
Zhang, Q. , Zhang, S. and Lin, Z. (2015) Spatial Segregation Limit of a Quasilinear Competition-Diffusion System. Applied Mathematics, 6, 1977-1987. doi: 10.4236/am.2015.612175.
References
[1]   Amann, H. (1990) Dynamic Theory of Quasilinear Parabolic Systems II. Reaction-Diffusion Systems, Differential Integral Equations, 3, 13-75.

[2]   Aronson, D.G., Crandall, M.G. and Peletier, L.A. (1982) Stabilization of Solutions oa a Degenerate Nonlinear Diffusion Problem. Nonlinear Analysis, 6, 1001-1022.
http://dx.doi.org/10.1016/0362-546X(82)90072-4

[3]   Constantin, A., Escher, J. and Yin, Z. (2004) Global Solutions for Quasilinear Parabolic Systems. Journal of Differential Equations, 197, 73-84.
http://dx.doi.org/10.1016/S0022-0396(03)00165-7

[4]   Ladyzenskaja, O.A., Solonnikov, V.A. and Ural’ceva, N.N. (1968) Linear and Quasi-Linear Equations of Parabolic Type. Transactions of the American Mathematical Society, Monographs 23, Providence.

[5]   Pao, C.V. (2007) Quasilinear Parabolic and Elliptic Equations with Nonlinear Boundary Conditions. Nonlinear Analysis, 66, 639-662.
http://dx.doi.org/10.1016/j.na.2005.12.007

[6]   Zhang, Q.Y. and Lin, Z.G. (2010) Periodic Solutions of Quasilinear Parabolic Systems with Nonlinear Boundary Conditions. Nonlinear Analysis, 72, 3429-3435.
http://dx.doi.org/10.1016/j.na.2009.12.026

[7]   Crooks, E.C.M., Dancer, E.N., Hilhorst, D., Mimura, M. and Ninomiya, H. (2004) Spatial Segregation Limit of a Competition Diffusion System with Dirichlet Boundary Conditions. Nonlinear Analysis: Real World Applications, 5, 645-665.
http://dx.doi.org/10.1016/j.nonrwa.2004.01.004

[8]   Dancer, E.N. and Du, Y.H. (1994) Competing Species Equations with Diffusion, Large Interactions, and Jumping Nonlinearities. Journal of Differential Equations, 114, 434-475.
http://dx.doi.org/10.1006/jdeq.1994.1156

[9]   Dancer, E.N., Hilhorst, D., Mimura, M. and Peletier, L.A. (1999) Spatial Segregation Limit of a Competition-Diffusion System. European Journal of Applied Mathematics, 10, 97-115.

[10]   Dancer, E.N., Wang, K. and Zhang, Z. (2012) The Limit Equation for the Gross-Pitaevskii Equations and S. Terracini’s Conjecture. Journal of Functional Analysis, 262, 1087-1131.
http://dx.doi.org/10.1016/j.jfa.2011.10.013

[11]   Namba, T. and Mimura, M. (1980) Spatial Distribution for Competing Populations. Journal of Theoretical Biology, 87, 795-814.
http://dx.doi.org/10.1016/0022-5193(80)90118-6

[12]   Shigesada, N., Kawasaki, K. and Teramoto, E. (1979) Spatial Segregation of Interacting Species. Journal of Theoretical Biology, 79, 83-99.
http://dx.doi.org/10.1016/0022-5193(79)90258-3

[13]   Wang, K.L. and Zhang, Z.T. (2010) Some New Results in Competing Systems with Many Species. Annales de l’Institut Henri Poincare (C) Non Linear Analysis, 27, 739-761.
http://dx.doi.org/10.1016/j.anihpc.2009.11.004

[14]   Wei, J.C. and Weth, T. (2008) Asymptotic Behaviour of Solutions of Planar Elliptic Systems with Strong Competition. Nonlinearity, 21, 305-317.
http://dx.doi.org/10.1088/0951-7715/21/2/006

[15]   Zhang, S., Zhou, L., Liu, Z.H. and Lin, Z.G. (2012) Spatial Segregation Limit of a Non-Autonomous Competition-Diffusion System. Journal of Mathematical Analysis and Applications, 389, 119-129.
http://dx.doi.org/10.1016/j.jmaa.2011.11.054

[16]   Caffarelli, L.A., Karakhanyan, A.L. and Lin, F.H. (2009) The Geometry of Solutions to a Segregation Problem for Nondivergence Systems. Journal of Fixed Point Theory and Applications, 5, 319-351.
http://dx.doi.org/10.1007/s11784-009-0110-0

[17]   Dancer, E.N., Wang, K.L. and Zhang, Z.T. (2011) Uniform Hölder Estimate for Singularly Perturbed Parabolic Systems of Bose-Einstein Condensates and Competing Species. Journal of Differential Equations, 251, 2737-2769.
http://dx.doi.org/10.1016/j.jde.2011.06.015

[18]   Conti, M., Terracini, S. and Verzini, G. (2005) Asymptotic Estimates for the Spatial Segregation of Competitive Systems. Advances in Mathematics, 195, 524-560.
http://dx.doi.org/10.1016/j.aim.2004.08.006

[19]   Zhang, S., Zhou, L. and Liu, Z.H. (2013) The Spatial Behavior of a Competition Diffusion Advection System with Strong Competition. Nonlinear Analysis: Real World Applications, 14, 976-989.
http://dx.doi.org/10.1016/j.nonrwa.2012.08.011

[20]   Chang, S.M., Lin, C.S., Lin, T.C. and Lin, W.W. (2004) Segregated Nodal Domains of Two-Dimensional Multispecies Bose-Einstein Condensates. Physica D: Nonlinear Phenomena, 196, 341-361.
http://dx.doi.org/10.1016/j.physd.2004.06.002

[21]   Noris, B., Tavares, H., Terracini, S. and Verzini, G. (2010) Uniform Hölder Bounds for Nonlinear Schrödinger Systems with Strong Competition. Communications on Pure and Applied Mathematics, 63, 267-302.

[22]   Soave, N. and Zilio, A. (2015) Uniform Bounds for Strongly Competing Systems: The Optimal Lipschitz Case. Archive for Rational Mechanics and Analysis, 218, 647-697.
http://dx.doi.org/10.1007/s00205-015-0867-9

[23]   Caffarelli, L.A. and Lin, F.H. (2008) Singularly Perturbed Elliptic Systems and Multi-Valued Harmonic Functions with Free Boundaries. Journal of the American Mathematical Society, 21, 847-862.
http://dx.doi.org/10.1090/S0894-0347-08-00593-6

[24]   Tavares, H. and Terracini, S. (2012) Regularity of the Nodal Set of Segregated Critical Configurations under a Weak Reflection Law. Calculus of Variations and Partial Differential Equations, 45, 273-317.
http://dx.doi.org/10.1007/s00526-011-0458-z

[25]   Caffarelli, L.A. and Lin, F. (2010) Analysis on the Junctions of Domain Walls. Discrete and Continuous Dynamical Systems, 28, 915-929.

[26]   Zhang, S. and Liu, Z.H. (2015) Singularities of the Nodal Set of Segregated Configurations. Calculus of Variations and Partial Differential Equations, 54, 2017-2037.
http://dx.doi.org/10.1007/s00526-015-0854-x

[27]   Evans, L.C. (1998) Partial Differential Equations. Graduate Studies in Mathematics, Vol. 19, American Mathematical Society, Providence.

[28]   Dibenedetto, E. (1993) Degenerate Parabolic Equations. Springer-Verlag, New York.
http://dx.doi.org/10.1007/978-1-4612-0895-2

[29]   Gilbarg, D. and Trudinger, N.S. (2001) Elliptic Partial Differential Equations of Second Order. 2nd Edition, Springer, New York.

 
 
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