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

Original Paper


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 Classication

35K92 45K05 

Key words and phrases

Nonlocal equation non-linear viscoelastic equation finite speed of propagation waiting time property heat flows with memory 


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  1. [1]
    G. Andrews, On the existence of solutions to the equation utt = uxxt+σ(ux)x, J. Differential Equations, 35 (1980), pp. 200–231.MathSciNetCrossRefGoogle Scholar
  2. [2]
    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
  3. [3]
    S. Antontsev, S. Shmarev, J. Simsen, and M. S. Simsen, On the evolution p-Laplacian with nonlocal memory, Nonlinear Anal., 134 (2016), pp. 31–54.MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    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
  5. [5]
    V. Barbu, Integro-differential equations in Hilbert spaces, An. Şti. Univ. “Al. I. Cuza” Iaşi Secţ. I a Mat. (N.S.), 19 (1973), pp. 365–383.MathSciNetMATHGoogle Scholar
  6. [6]
    T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuss, and J. Valero, Global attractor for a non-autonomous integro-differential equation in materials with memory, Nonlinear Anal., 73 (2010), pp. 183–201.MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    V. V. Chepyzhov and A. Miranville, On trajectory and global attractors for semilinear heat equations with fading memory, Indiana Univ. Math. J., 55 (2006), pp. 119–167.MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    M. Conti, E. M. Marchini, and V. Pata, Reaction-diffusion with memory in the minimal state framework, Trans. Amer. Math. Soc., 366 (2014), pp. 4969–4986.MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    M. G. Crandall, S.-O. Londen, and J. A. Nohel, An abstract nonlinear Volterra integrodifferential equation, J. Math. Anal. Appl., 64 (1978), pp. 701–735.MathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    C. Dafermos, The mixed initial-boundary value problem for the equations of nonlinear one-dimensional viscoelasticity, J. Differential Equations, 6 (1969), pp. 71–86.MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    J.I. Diaz and H. Gomez, On the interfaces for some integrodifferential evolution equations: the qualitative and numerical approaches. In preparationGoogle Scholar
  12. [12]
    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
  13. [13]
    L. Du and C. Mu, Global existence and blow-up analysis to a degenerate reaction-diffusion system with nonlinear memory, Nonlinear Anal. Real World Appl., 9 (2008), pp. 303–315.MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    -, Global existence and blow-up analysis to a degenerate reaction-diffusion system with nonlinear memory, Nonlinear Anal. Real World Appl., 9 (2008), pp. 303–315.MathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    H. Engler, Weak solutions of a class of quasilinear hyperbolic integro-differential equations describing viscoelastic materials, Arch. Rational Mech. Anal., 113 (1990), pp. 1–38.MathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    Z. B. Fang and J. Zhang, Global existence and blow-up of solutions for p-Laplacian evolution equation with nonlinear memory term and nonlocal boundary condition, Bound. Value Probl., (2014), 2014:8, 17.MathSciNetMATHGoogle Scholar
  17. [17]
    J. Greenberg, R. MacCamy, and V. Mizei, On the existence, uniqueness and stability of the equation Ã′(ux)uxx+λuxtx=ρ0utt, J. Math. Mech., 17 (1968), pp. 707–728.MathSciNetGoogle Scholar
  18. [18]
    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
  19. [19]
    A. I. Kozhanov, N. A. Lar’kin, and N. N. Yanenko, A mixed problem for a class of third-order equations, Sibirsk. Mat. Zh., 22 (1981), no.6, 81–86, 225.MathSciNetMATHGoogle Scholar
  20. [20]
    C. Li, L. Qiu, and Z. B. Fang, General decay rate estimates for a semilinear parabolic equation with memory term and mixed boundary condition, Bound. Value Probl., (2014), 2014:197, 11.MathSciNetMATHGoogle Scholar
  21. [21]
    Y. Li and C. Xie, Blow-up for semilinear parabolic equations with nonlinear memory, Z. Angew. Math. Phys., 55 (2004), pp. 15–27.MathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    G. Liu and H. Chen, Global and blow-up of solutions for a quasilinear parabolic system with viscoelastic and source terms, Math. Methods Appl. Sci., 37 (2014), pp. 148–156.MathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    R. C. MacCamy, Stability theorems for a class of functional differential equations, SIAM J. Appl. Math., 30 (1976), pp. 557–576.MathSciNetCrossRefMATHGoogle Scholar
  24. [24]
    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
  25. [25]
    -, 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
  26. [26]
    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
  27. [27]
    -, Nonlinear Volterra equations for heat flow in materials with memory, in Integral and functional differential equations (Proc. Conf., West Virginia Univ., Morgantown, W. Va., 1979), vol. 67 of Lecture Notes in Pure and Appl. Math., Dekker, New York, 1981, pp. 3–82.MathSciNetGoogle Scholar
  28. [28]
    J. Pruss; Evolutionary integral equations and applications, Volume 87 of Monographs in Mathematics, Birkhäuser Verlag, Basel, 1993.CrossRefMATHGoogle Scholar
  29. [29]
    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
  30. [30]
    U. Stefanelli, On some nonlocal evolution equations in Banach spaces, J. Evol. Equ. 4(2004), pp. 1–26.MathSciNetCrossRefMATHGoogle Scholar
  31. [31]
    Y. Sun, G. Li, and W. Liu, General decay of solutions for a singular nonlocal viscoelastic problem with nonlinear damping and source, J. Comput. Anal. Appl., 16 (2014), pp. 50–55.MathSciNetMATHGoogle Scholar
  32. [32]
    J. Yong and X. Zhang, Heat equations with memory, Nonlinear Analysis 63 (2005) e99 e108.Google Scholar
  33. [33]
    K. Yoshida, Energy inequalities and finite propagation speed of the Cauchy problem for hyperbolic equations with constantly multiple characteristics, Proc. Japan Acad. 50 (1974) 561–565.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Orthogonal Publishing and Springer International Publishing 2016

Authors and Affiliations

  1. 1.M.A. Lavrentyev Institute of Hydrodynamics SB RASNovosibirskRussia
  2. 2.Novosibirsk State UniversityNovosibirskRussia
  3. 3.CMAF-CIOUniversity of LisbonLisbonPortugal
  4. 4.Instituto de Matemática InterdisciplinarUniversidad Complutense de MadridSpain

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