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On Phase-Separation Models: Asymptotics and Qualitative Properties

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Abstract

In this paper we study bound state solutions of a class of two-component nonlinear elliptic systems with a large parameter tending to infinity. The large parameter giving strong intercomponent repulsion induces phase separation and forms segregated nodal domains divided by an interface. To obtain the profile of bound state solutions near the interface, we prove the uniform Lipschitz continuity of bound state solutions when the spatial dimension is N = 1. Furthermore, we show that the limiting nonlinear elliptic system that arises has unbounded solutions with symmetry and monotonicity. These unbounded solutions are useful for rigorously deriving the asymptotic expansion of the minimizing energy which is consistent with the hypothesis of Du and Zhang (Discontin Dynam Sys, 2012). When the spatial dimension is N = 2, we establish the De Giorgi type conjecture for the blow-up nonlinear elliptic system under suitable conditions at infinity on bound state solutions. These results naturally lead us to formulate De Giorgi type conjectures for these types of systems in higher dimensions.

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Authors and Affiliations

  1. Ecole des hautes etudes en sciences sociales, CAMS, 54, boulevard Raspail, 75006, Paris, France

    Henri Berestycki

  2. Department of Mathematics, National Center for Theoretical Sciences at Taipei, National Taiwan University, Taipei, 10617, Taiwan

    Tai-Chia Lin

  3. Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong

    Juncheng Wei

  4. Department of Mathematics, East China Normal University, Shanghai, 200241, China

    Chunyi Zhao

  5. Department of Mathematics, Taida Institute of Mathematical Sciences (TIMS), National Taiwan University, Taipei, 10617, Taiwan

    Chunyi Zhao

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  1. Henri Berestycki
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  2. Tai-Chia Lin
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  3. Juncheng Wei
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  4. Chunyi Zhao
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Correspondence to Henri Berestycki.

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Communicated by F. Lin

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Berestycki, H., Lin, TC., Wei, J. et al. On Phase-Separation Models: Asymptotics and Qualitative Properties. Arch Rational Mech Anal 208, 163–200 (2013). https://doi.org/10.1007/s00205-012-0595-3

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  • DOI: https://doi.org/10.1007/s00205-012-0595-3

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