# American Institute of Mathematical Sciences

January  2015, 35(1): 483-512. doi: 10.3934/dcds.2015.35.483

## The singular limit problem in a phase separation model with different diffusion rates $^*$

 1 Wuhan Institute of Physics and Mathematics, The Chinese Academy of Sciences, Wuhan 430071, China

Received  August 2013 Revised  June 2014 Published  August 2014

In this paper we study the singularly perturbed parabolic system of competing species. This problem exhibit a phase separation" phenomena when the interaction between different species is very strong. We are concerned with the case where different species may have different diffusion rates. We identify its singular limit with the heat flow (i.e. gradient flow) of harmonic maps into a metric space with non-positive curvature, by establishing the system of differential inequalities satisfied by this heat flow and uniqueness of the solution to the corresponding initial-boundary value problem.
Citation: Kelei Wang. The singular limit problem in a phase separation model with different diffusion rates $^*$. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 483-512. doi: 10.3934/dcds.2015.35.483
##### References:
 [1] H. Brezis and A. Ponce, Kato's inequality when $\Delta u$ is a measure, C. R. Math. Acad. Sci. Paris, 338 (2004), 599-604. doi: 10.1016/j.crma.2003.12.032. [2] L. Caffarelli, A. Karakhanyan and F. Lin, The geometry of solutions to a segregation problem for non-divergence systems, J. Fixed Point Theory Appl., 5 (2009), 319-351. doi: 10.1007/s11784-009-0110-0. [3] L. Caffarelli and F. Lin, Singularly perturbed elliptic systems and multi-valued harmonic functions with free boundaries, Journal of the American Mathematical Society, 21 (2008), 847-862. doi: 10.1090/S0894-0347-08-00593-6. [4] L. Caffarelli and F. Lin, Nonlocal heat flows preserving the $L^2$ energy, Discrete and Continuous Dynamical Systems (DCDS-A), 23 (2009), 49-64. doi: 10.3934/dcds.2009.23.49. [5] L. Caffarelli and S. Salsa, A geometric Approach to Free Boundary Problems, Graduate Studies in Mathematics, 68, American Mathematical Society, Providence, RI, 2005. [6] J. Cannon and C. Hill, On the movement of a chemical reaction interface, Indiana Univ. Math. J., 20 (1970), 429-454. doi: 10.1512/iumj.1971.20.20037. [7] M. Conti, S. Terracini and G. Verzini, A variational problem for the spatial segregation of reaction diffusion systems, Indiana Univ. Math. J., 54 (2005), 779-815. doi: 10.1512/iumj.2005.54.2506. [8] M. Conti, S. Terracini and G. Verzini, Asymptotic estimates for the spatial segregation of competitive systems, Adv. Math., 195 (2005), 524-560. doi: 10.1016/j.aim.2004.08.006. [9] E. N. Dancer, D. Hilhorst, M. Mimura and L. A. Peletier, Spatial segregation limit of a competition-diffusion system, European J. Appl. Math., 10 (1999), 97-115. doi: 10.1017/S0956792598003660. [10] E. N. Dancer, K. Wang and Z. Zhang, Dynamics of strongly competing systems with many species, Trans. Amer. Math. Soc., 364 (2012), 961-1005. doi: 10.1090/S0002-9947-2011-05488-7. [11] E. N. Dancer, K. Wang and Z. Zhang, Uniform Hölder estimate for singularly perturbed parabolic systems of Bose-Einstein condensates and competing species, Journal of Differential Equations, 251 (2011), 2737-2769. doi: 10.1016/j.jde.2011.06.015. [12] E. N. Dancer, K. Wang and Z. Zhang, The limit equation for the Gross-Pitaevskii equations and S. Terracini's conjecture, Journal of Functional Analysis, 262 (2012), 1087-1131. doi: 10.1016/j.jfa.2011.10.013. [13] E. N. Dancer, K. Wang and Z. Zhang, Addendum to "The limit equation for the Gross-Pitaevskii equations and S. Terracini's conjecture" [J. Funct. Anal., 262 (2012), 1087-1131], Journal of Functional Analysis, 264 (2013), 1125-1129. doi: 10.1016/j.jfa.2012.10.009. [14] E. N. Dancer and Z. Zhang, Dynamics of Lotka-Volterra competition systems with large interaction, Journal of Differential Equations, 182 (2002), 470-489. doi: 10.1006/jdeq.2001.4102. [15] E. De Giorgi, New problems on minimizing movements, in Boundary Value Problems for PDE and Applications (eds. C. Baiochhi and J.L. Lions), RMA Res. Notes Appl. Math., 29, Masson, Paris, 1993, 81-98. [16] L. Evans, A chemical diffusion-reaction free boundary problem, Nonlinear Anal., 6 (1982), 455-466. doi: 10.1016/0362-546X(82)90059-1. [17] N. Gigli, On the heat flow on metric measure spaces: Existence, uniqueness and stability, Calculus of Variations and Partial Differential Equations, 39 (2010), 101-120. doi: 10.1007/s00526-009-0303-9. [18] M. Gromov and R. Schoen, Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one, Publications Mathématiques de L'IHÉS, 76 (1992), 165-246. [19] W. Littman, G. Stampacchia and H. Weinberger, Regular points for elliptic equations with discontinuous coefficients, Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, 17 (1963), 43-77. [20] F. Lin and X. Yang, Geometric Measure Theory: An Introduction, Advanced Mathematics (Beijing/Boston), Science Press, Beijing; International Press, Boston, 2002. [21] U. Mayer, Gradient flows on nonpositively curved metric spaces and harmonic maps, Commun. Anal. Geom., 6 (1998), 199-253. [22] B. Noris, H. Tavares, S. Terracini and G. Verzini, Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition, Comm. Pure Appl. Math., 63 (2010), 267-302. [23] W. Rudin, Real and Complex Analysis, Tata McGraw-Hill Education, 2006. [24] K. Wang and Z. Zhang, Some new results in competing systems with many species, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 27 (2010), 739-761. doi: 10.1016/j.anihpc.2009.11.004.

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##### References:
 [1] H. Brezis and A. Ponce, Kato's inequality when $\Delta u$ is a measure, C. R. Math. Acad. Sci. Paris, 338 (2004), 599-604. doi: 10.1016/j.crma.2003.12.032. [2] L. Caffarelli, A. Karakhanyan and F. Lin, The geometry of solutions to a segregation problem for non-divergence systems, J. Fixed Point Theory Appl., 5 (2009), 319-351. doi: 10.1007/s11784-009-0110-0. [3] L. Caffarelli and F. Lin, Singularly perturbed elliptic systems and multi-valued harmonic functions with free boundaries, Journal of the American Mathematical Society, 21 (2008), 847-862. doi: 10.1090/S0894-0347-08-00593-6. [4] L. Caffarelli and F. Lin, Nonlocal heat flows preserving the $L^2$ energy, Discrete and Continuous Dynamical Systems (DCDS-A), 23 (2009), 49-64. doi: 10.3934/dcds.2009.23.49. [5] L. Caffarelli and S. Salsa, A geometric Approach to Free Boundary Problems, Graduate Studies in Mathematics, 68, American Mathematical Society, Providence, RI, 2005. [6] J. Cannon and C. Hill, On the movement of a chemical reaction interface, Indiana Univ. Math. J., 20 (1970), 429-454. doi: 10.1512/iumj.1971.20.20037. [7] M. Conti, S. Terracini and G. Verzini, A variational problem for the spatial segregation of reaction diffusion systems, Indiana Univ. Math. J., 54 (2005), 779-815. doi: 10.1512/iumj.2005.54.2506. [8] M. Conti, S. Terracini and G. Verzini, Asymptotic estimates for the spatial segregation of competitive systems, Adv. Math., 195 (2005), 524-560. doi: 10.1016/j.aim.2004.08.006. [9] E. N. Dancer, D. Hilhorst, M. Mimura and L. A. Peletier, Spatial segregation limit of a competition-diffusion system, European J. Appl. Math., 10 (1999), 97-115. doi: 10.1017/S0956792598003660. [10] E. N. Dancer, K. Wang and Z. Zhang, Dynamics of strongly competing systems with many species, Trans. Amer. Math. Soc., 364 (2012), 961-1005. doi: 10.1090/S0002-9947-2011-05488-7. [11] E. N. Dancer, K. Wang and Z. Zhang, Uniform Hölder estimate for singularly perturbed parabolic systems of Bose-Einstein condensates and competing species, Journal of Differential Equations, 251 (2011), 2737-2769. doi: 10.1016/j.jde.2011.06.015. [12] E. N. Dancer, K. Wang and Z. Zhang, The limit equation for the Gross-Pitaevskii equations and S. Terracini's conjecture, Journal of Functional Analysis, 262 (2012), 1087-1131. doi: 10.1016/j.jfa.2011.10.013. [13] E. N. Dancer, K. Wang and Z. Zhang, Addendum to "The limit equation for the Gross-Pitaevskii equations and S. Terracini's conjecture" [J. Funct. Anal., 262 (2012), 1087-1131], Journal of Functional Analysis, 264 (2013), 1125-1129. doi: 10.1016/j.jfa.2012.10.009. [14] E. N. Dancer and Z. Zhang, Dynamics of Lotka-Volterra competition systems with large interaction, Journal of Differential Equations, 182 (2002), 470-489. doi: 10.1006/jdeq.2001.4102. [15] E. De Giorgi, New problems on minimizing movements, in Boundary Value Problems for PDE and Applications (eds. C. Baiochhi and J.L. Lions), RMA Res. Notes Appl. Math., 29, Masson, Paris, 1993, 81-98. [16] L. Evans, A chemical diffusion-reaction free boundary problem, Nonlinear Anal., 6 (1982), 455-466. doi: 10.1016/0362-546X(82)90059-1. [17] N. Gigli, On the heat flow on metric measure spaces: Existence, uniqueness and stability, Calculus of Variations and Partial Differential Equations, 39 (2010), 101-120. doi: 10.1007/s00526-009-0303-9. [18] M. Gromov and R. Schoen, Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one, Publications Mathématiques de L'IHÉS, 76 (1992), 165-246. [19] W. Littman, G. Stampacchia and H. Weinberger, Regular points for elliptic equations with discontinuous coefficients, Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, 17 (1963), 43-77. [20] F. Lin and X. Yang, Geometric Measure Theory: An Introduction, Advanced Mathematics (Beijing/Boston), Science Press, Beijing; International Press, Boston, 2002. [21] U. Mayer, Gradient flows on nonpositively curved metric spaces and harmonic maps, Commun. Anal. Geom., 6 (1998), 199-253. [22] B. Noris, H. Tavares, S. Terracini and G. Verzini, Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition, Comm. Pure Appl. Math., 63 (2010), 267-302. [23] W. Rudin, Real and Complex Analysis, Tata McGraw-Hill Education, 2006. [24] K. Wang and Z. Zhang, Some new results in competing systems with many species, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 27 (2010), 739-761. doi: 10.1016/j.anihpc.2009.11.004.
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