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Anomalous Diffusion in Polymers: Long-Time Behaviour

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Infinite Dimensional Dynamical Systems

Part of the book series: Fields Institute Communications ((FIC,volume 64))

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Abstract

We study the Dirichlet boundary value problem for viscoelastic diffusion in polymers. We show that its weak solutions generate a dissipative semiflow. We construct the minimal trajectory attractor and the global attractor for this problem.

Mathematics Subject Classification 2010(2010): Primary 35B41, 35D99; Secondary 76R50, 82D60

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Notes

  1. 1.

    Formula (4) describes the following peculiarities of the processes under consideration. The polymer network in the glassy state (low concentration area) is severely entangled, so β is approximately equal to some small βG. In the high concentration areas the system is in the rubbery state: the network disentangles, so the relaxation time is small, and its inverse is close to βR > βG. The glass-rubber phase transition occurs near a certain concentration uRG. However, we assume that β also depends on stress, cf. [2, 11, 26].

  2. 2.

    Trajectory attractors for problems with uniqueness were investigated in [6] only as an intermediate step on the way to usual global attractors of semigroups.

  3. 3.

    The case μ = 0 (“the Maxwell model” [9, 30]) is admissible as well.

  4. 4.

    e.g. in form (4).

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Acknowledgements

The work was partially supported by RFBR.

Received 9/8/2009; Accepted 6/3/2010

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Authors and Affiliations

  1. Faculty of Mathematics, Voronezh State University, Universitetskaya Pl. 1, 394006, Voronezh, Russia

    Dmitry A. Vorotnikov

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  1. Dmitry A. Vorotnikov
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Correspondence to Dmitry A. Vorotnikov .

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Editors and Affiliations

  1. Division of Applied Mathematics, Brown University, Providence, RI, USA

    John Mallet-Paret

  2. Department of Mathematics and Statistics, York University, Toronto, ON, Canada

    Jianhong Wu  & Huaiping Zhu  & 

  3. Georgia Institute of Technology, School of Mathematics, Atlanta, GA, USA

    Yingfei Yi

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Vorotnikov, D.A. (2013). Anomalous Diffusion in Polymers: Long-Time Behaviour. In: Mallet-Paret, J., Wu, J., Yi, Y., Zhu, H. (eds) Infinite Dimensional Dynamical Systems. Fields Institute Communications, vol 64. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4523-4_19

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