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Finite Speed of Propagation and Waiting Time for Local Solutions of Degenerate Equations in Viscoelastic Media or Heat Flows with Memory

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Abstract

The finite speed of propagation (FSP) was established for certain materials in the 70’s by the American school (Gurtin, Dafermos, Nohel, etc.) for the special case of the presence of memory effects. A different approach can be applied by the construction of suitable super and sub-solutions (Crandall, Nohel, Díaz and Gomez, etc.). In this paper we present an alternative method to prove (FSP) which only uses some energy estimates and without any information coming from the characteristics analysis. The waiting time property is proved for the first time in the literature for this class of nonlocal equations.

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Authors and Affiliations

  1. M.A. Lavrentyev Institute of Hydrodynamics SB RAS, Novosibirsk, Russia

    S. N. Antontsev

  2. Novosibirsk State University, Novosibirsk, Russia

    S. N. Antontsev

  3. CMAF-CIO, University of Lisbon, Lisbon, Portugal

    S. N. Antontsev

  4. Instituto de Matemática Interdisciplinar, Universidad Complutense de Madrid, Spain

    J. I. Díaz

Authors
  1. S. N. Antontsev
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  2. J. I. Díaz
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Correspondence to S. N. Antontsev.

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Dedicated to Professor David Kinderlehrer on occasion of his 75th birthday.

The research of SNA was partially supported by the Project UID/MAT/04561/2013 of the Portuguese Foundation for Science and Techology (FCT), Portugal and by the Grant No.15-11-20019 of Russian Science Foundation, Russia. The research of JID was partially supported by the project Ref.MTM2014-57113-P of the DGISPI (Spain) and the Research Group MOMAT (Ref. 910480) supported by the Universidad Complutense de Madrid.

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Antontsev, S.N., Díaz, J.I. Finite Speed of Propagation and Waiting Time for Local Solutions of Degenerate Equations in Viscoelastic Media or Heat Flows with Memory. J Elliptic Parabol Equ 2, 207–216 (2016). https://doi.org/10.1007/BF03377402

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