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Shadow Limit Using Renormalization Group Method and Center Manifold Method

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Abstract

We study a shadow limit (the infinite diffusion coefficient-limit) of a system of ODEs coupled with a semilinear heat equation in a bounded domain with Neumann boundary conditions. In the literature, it was established formally that in the limit, the original semilinear heat equation reduces to an ODE involving the space averages of the solution to the semilinear heat equation and of the nonlinearity. It is coupled with the original system of ODEs for every space point x. We present derivation of the limit using the renormalization group (RG) and the center manifold approaches. The RG approach provides also further approximating expansion terms. The error estimate in the terms of the inverse of the diffusion coefficient is obtained for the finite time intervals. For the infinite times, the center manifolds for the starting problem and for its shadow limit approximation are compared and it is proved that their distance is of the order of the inverse of the diffusion coefficient.

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Acknowledgments

Andro Mikelić would like to express his thanks to The Heidelberg Graduates School HGS MathComp, IWR, Universität Heidelberg, for partially supporting his post-Romberg professorship research visits in 2014-2015. Anna Marciniak-Czochra acknowledges the support of the Emmy Noether Programme of the German Research Council (DFG).

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Authors and Affiliations

  1. Institute of Applied Mathematics, IWR and BIOQUANT, University of Heidelberg, Im Neuenheimer Feld 205, 69120, Heidelberg, Germany

    Anna Marciniak-Czochra

  2. Institut Camille Jordan, Université de Lyon, CNRS UMR 5208, Université Lyon 1, 43, blvd. du 11 novembre 1918, 69622, Villeurbanne Cedex, France

    Andro Mikelić

Authors
  1. Anna Marciniak-Czochra
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  2. Andro Mikelić
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Correspondence to Andro Mikelić.

Additional information

This contribution is dedicated to Professor Willi Jäger on his 75th birthday. Multiscale analysis and biological applications are two subjects of focus in Willi’s research over his professional life.

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Marciniak-Czochra, A., Mikelić, A. Shadow Limit Using Renormalization Group Method and Center Manifold Method. Vietnam J. Math. 45, 103–125 (2017). https://doi.org/10.1007/s10013-016-0199-6

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  • DOI: https://doi.org/10.1007/s10013-016-0199-6

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