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Dynamics of interfaces in a scalar parabolic equation with variable diffusion coefficients

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Abstract

Consider the equationu t = ε2 div(D(x)∇u) +f(u; ε) in ℝn, whereD(x) is a positive functionx ∈ ℝn,f is the derivative of a bistable potential, and ε>0 is a small parameter. Let Γ(T),T∈[0,T 0], be a one-parameter family of smooth hypersurfaces which move with the time scaleT = ε2 t according to a certain generalized mean curvature flow. It is shown that, if the initial data have an interface which is close to Γ(0), then the interface remains close to Γ(ε2 t fort ∈ [0,T 0/ε2]. Moreover, ifT 0=∞ and Γ(T) converges to a stable stationary hypersurface asT→∞, then the interface remains close to Γ(ε2 t) for allt≥0.

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Authors and Affiliations

  1. Graduate School of Integrated Science, Yokohama City University, 236, Yokohama, Japan

    Shin-Ichiro Ei

  2. Department of Mathematics, Faculty of Humanities and Social Sciences, Iwate University, 020, Morioka, Japan

    Masato Iida

  3. Graduate School of Mathematical Sciences, The University of Tokyo, Meguro-ku, 153, Tokyo, Japan

    Eiji Yanagida

Authors
  1. Shin-Ichiro Ei
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  2. Masato Iida
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  3. Eiji Yanagida
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Dedicated to Professor Kyûya Masuda on his sixtieth birthday

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Ei, SI., Iida, M. & Yanagida, E. Dynamics of interfaces in a scalar parabolic equation with variable diffusion coefficients. Japan J. Indust. Appl. Math. 14, 1–23 (1997). https://doi.org/10.1007/BF03167306

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  • DOI: https://doi.org/10.1007/BF03167306

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