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

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.

This is a preview of subscription content, log in to check access.

References

  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.

    MathSciNet  Article  Google 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.

    MathSciNet  Article  MATH  Google 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.

    MathSciNet  MATH  Google 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.

    MathSciNet  Article  MATH  Google 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.

    MathSciNet  Article  MATH  Google 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.

    MathSciNet  Article  MATH  Google 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.

    MathSciNet  Article  MATH  Google 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.

    MathSciNet  Article  MATH  Google Scholar 

  11. [11]

    J.I. Diaz and H. Gomez, On the interfaces for some integrodifferential evolution equations: the qualitative and numerical approaches. In preparation

  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. 1–13

    Google 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.

    MathSciNet  Article  MATH  Google 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.

    MathSciNet  Article  MATH  Google 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.

    MathSciNet  Article  MATH  Google 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.

    MathSciNet  MATH  Google 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.

    MathSciNet  Google 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.

  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.

    MathSciNet  MATH  Google 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.

    MathSciNet  MATH  Google 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.

    MathSciNet  Article  MATH  Google 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.

    MathSciNet  Article  MATH  Google Scholar 

  23. [23]

    R. C. MacCamy, Stability theorems for a class of functional differential equations, SIAM J. Appl. Math., 30 (1976), pp. 557–576.

    MathSciNet  Article  MATH  Google 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.

    MathSciNet  Google 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.

    MathSciNet  Google Scholar 

  28. [28]

    J. Pruss; Evolutionary integral equations and applications, Volume 87 of Monographs in Mathematics, Birkhäuser Verlag, Basel, 1993.

    Google 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.

    MathSciNet  Article  MATH  Google 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.

    MathSciNet  MATH  Google Scholar 

  32. [32]

    J. Yong and X. Zhang, Heat equations with memory, Nonlinear Analysis 63 (2005) e99 e108.

  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.

    MathSciNet  Article  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to S. N. Antontsev.

Additional information

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.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

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