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Stabilization of Solutions to a FitzHugh-Nagumo Type System

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Abstract

We consider a bistable reaction-diffusion system arising in the theory of phase transitions; it appears in several physical contexts such as thin magnetic films and the microphase separation in diblock copolymer melts. Mathematically it takes the form of an Allen-Cahn equation coupled to an elliptic equation. This system possesses a Lyapunov functional which represents the Gibbs free energy of the phase separation problem. We study the large time behavior of the solution orbits, and use the fact that the problem has a gradient structure to prove their stabilization by means of a version of Łojasiewicz inequality.

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Authors and Affiliations

  1. CNRS and Laboratoire de Mathématiques, Université de Paris-Sud 11, Bât. 425, 91405, Orsay, France

    Danielle Hilhorst

  2. Institute of Applied Mathematics and Mechanics, The University of Warsaw, ul. Banacha 2, 02-097, Warszawa, Poland

    Piotr Rybka

Authors
  1. Danielle Hilhorst
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  2. Piotr Rybka
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Correspondence to Danielle Hilhorst.

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Hilhorst, D., Rybka, P. Stabilization of Solutions to a FitzHugh-Nagumo Type System. J Stat Phys 138, 291–304 (2010). https://doi.org/10.1007/s10955-009-9886-y

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  • DOI: https://doi.org/10.1007/s10955-009-9886-y

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