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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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.
2010 Mathematics Subject Classication35K92 45K05
Key words and phrasesNonlocal equation non-linear viscoelastic equation finite speed of propagation waiting time property heat flows with memory
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- S. Antontsev and S. Shmarev, Evolution PDEs with nonstandard growth conditions: Existence, uniqueness, localization, blow-up, vol. 4 of Atlantis Studies in Differential Equations, Atlantis Press, Paris, 2015.Google Scholar
- S. N. Antontsev, J. I. Díaz, and S. Shmarev, Energy methods for free boundary problems, Progress in Nonlinear Differential Equations and their Applications, 48, Birkhäuser Boston, Inc., Boston, MA, 2002. Applications to nonlinear PDEs and fluid mechanics.Google Scholar
- J.I. Diaz and H. Gomez, On the interfaces for some integrodifferential evolution equations: the qualitative and numerical approaches. In preparationGoogle Scholar
- J. I. Díaz, T. Pirantozzi, L. Vázquez, On the finite time extinction phenomenon for some nonlinear fractional evolution equations, Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 239, pp. 1–13Google Scholar
- D. Kinderlehrer and G. Stampacchia, An introduction to variational inequalities and their applications. Reprint of the 1980 original. Classics in Applied Mathematics, 31. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000.Google Scholar
- S. A. Messaoudi, Blow-up of solutions of a semilinear heat equation with a memory term, Abstr. Appl. Anal., (2005), pp. 87–94.Google Scholar
- -, Blow-up of solutions of a semilinear heat equation with a visco-elastic term, in Nonlinear elliptic and parabolic problems, vol. 64 of Progr. Nonlinear Differential Equations Appl., Birkhäuser, Basel, 2005, pp. 351–356.Google Scholar
- J. A. Nohel, A nonlinear hyperbolic Volterra equation occurring in viscoelastic motion, in Transactions of the Twenty-Fifth Conference of Army Mathematicians (Johns Hopkins Univ., Baltimore, Md., 1979), vol. 1 of ARO Rep. 80, U. S. Army Res. Office, Research Triangle Park, N.C., 1980, pp. 177–184.MathSciNetGoogle Scholar
- M. Renardy, W.J. Hrusa and J.A. Nohel, Mathematical Problems in Viscoelasticity, Pitman Monographs and Surveys in Pure and Applied Mathematics 35, Longman 1987.Google Scholar
- J. Yong and X. Zhang, Heat equations with memory, Nonlinear Analysis 63 (2005) e99 e108.Google Scholar