Coexistence of Infinitely Many Stable Solutions to Reaction Diffusion Systems in the Singular Limit

  • Yasumasa Nishiura
Part of the Dynamics Reported book series (DYNAMICS, volume 3)


Recognition stems from realization of the separation boundary between two different physical or chemicalstates. In other words we can observe natural phenomena through the emergence and evolution of theinterface between these states as in solidification, combustion, chemical reaction, and biological patterns. The interface studied here results from the balance between two opposing tendencies: adiffusive effect and a (physical or chemical)separation kinetics built in the system. The former attempts to smooth out the inhomogeneity as in the heat equation, and the latter drives the system to one or the other pure state such as solid or liquid (see, for instance, Fife [19] for details). Turing’s contribution [58] is one of the pioneering works related to the onset of spatial patterns through a cooperative work of diffusion and separation kinetics. Besides the existence of these two tendencies, another key ingredient to produce interesting interfacial patterns is the differences of the strength of the above mixing and unmixing effects among species involved in the system. In fact, reaction diffusion systems for two componentsu and v, which are the main concern in this paper, can be classified formally as
  1. (i)

    There is a difference in the diffusion rates of u and v;

  2. (ii)

    There is a difference in the reaction rates of u and v;

  3. (ii)

    There are differences in the diffusion and reaction rates of u and v, i.e., a combination of (i) and (ii).



Stable Solution Reaction Diffusion System Principal Eigenvalue Singular Limit Slow Manifold 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • Yasumasa Nishiura
    • 1
  1. 1.Division of Mathematics and Informatics, Faculty of Integrated Arts and SciencesHiroshima UniversityHigashi-HiroshimaJapan

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