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

Original Paper

First Online:

## 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.

## 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## Preview

Unable to display preview. Download preview PDF.

## References

- [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]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]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.MathSciNetCrossRefzbMATHGoogle Scholar - [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]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.MathSciNetzbMATHGoogle Scholar - [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.MathSciNetCrossRefzbMATHGoogle Scholar - [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.MathSciNetCrossRefzbMATHGoogle Scholar - [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.MathSciNetCrossRefzbMATHGoogle Scholar - [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.MathSciNetCrossRefzbMATHGoogle Scholar - [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.MathSciNetCrossRefzbMATHGoogle Scholar - [11]J.I. Diaz and H. Gomez, On the interfaces for some integrodifferential evolution equations: the qualitative and numerical approaches. In preparationGoogle Scholar
- [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]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.MathSciNetCrossRefzbMATHGoogle Scholar - [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.MathSciNetCrossRefzbMATHGoogle Scholar - [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.MathSciNetCrossRefzbMATHGoogle Scholar - [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.MathSciNetzbMATHGoogle Scholar - [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]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]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.MathSciNetzbMATHGoogle Scholar - [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.MathSciNetzbMATHGoogle Scholar - [21]Y. Li and C. Xie,
*Blow-up for semilinear parabolic equations with nonlinear memory*, Z. Angew. Math. Phys., 55 (2004), pp. 15–27.MathSciNetCrossRefzbMATHGoogle Scholar - [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.MathSciNetCrossRefzbMATHGoogle Scholar - [23]R. C. MacCamy,
*Stability theorems for a class of functional differential equations*, SIAM J. Appl. Math., 30 (1976), pp. 557–576.MathSciNetCrossRefzbMATHGoogle Scholar - [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]-,
*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]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]-,
*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]J. Pruss; Evolutionary integral equations and applications, Volume 87 of Monographs in Mathematics, Birkhäuser Verlag, Basel, 1993.CrossRefzbMATHGoogle Scholar
- [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]U. Stefanelli,
*On some nonlocal evolution equations in Banach spaces*, J. Evol. Equ. 4(2004), pp. 1–26.MathSciNetCrossRefzbMATHGoogle Scholar - [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.MathSciNetzbMATHGoogle Scholar - [32]J. Yong and X. Zhang, Heat equations with memory,
*Nonlinear Analysis*63 (2005) e99 e108.Google Scholar - [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.MathSciNetCrossRefzbMATHGoogle Scholar

## Copyright information

© Orthogonal Publishing and Springer International Publishing 2016