Communications in Mathematical Physics

, Volume 146, Issue 3, pp 585–609 | Cite as

Convergence to diffusion waves of solutions to nonlinear viscoelastic model with fading memory

  • Yanni Zeng
Article

Abstract

We study the large time behavior inL2 of solutions to a model for the motion of an unbounded, homogeneous, viscoelastic bar with fading memory. Decay rates for the solutions are obtained under the assumption that the initial data and histories are smooth and small. Moreover, convergence of the solutions to diffusion waves, which are solutions of Burgers equations, is proved and rates are obtained. Our method is based on the study of properties of the solutions to the linearized system in the Fourier space.

Keywords

Neural Network Fourier Statistical Physic Complex System Initial Data 

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References

  1. 1.
    Liu, T.-P.: Nonlinear Waves for Viscoelasticity with Fading Memory. J. Differ. Eqs.76, 26–46 (1988)Google Scholar
  2. 2.
    Kawashima, S.: Large-time Behavior of Solutions to Hyperbolic-Parabolic Systems of Conservation Laws and Applications. Proc. Roy. Soc. Edinburgh106 A, 169–194 (1987)Google Scholar
  3. 3.
    Liu, T.-P.: Interactions of Nonlinear Hyperbolic Waves. In: Liu, F.-C., Liu, T.-P. (eds.) Nonlinear Analysis, pp. 171–184. Singapore: World Scientific 1991Google Scholar
  4. 4.
    Chern, I.-L., Liu, T.-P.: Convergence to Diffusion Waves of Solutions for Viscous Conservation Laws. Commun. Math. Phys.110, 503–517 (1987),120, 525–527 (1989)Google Scholar
  5. 5.
    Hrusa, W., Nohel, J.: The Cauchy Problem in One-Dimensional Nonlinear Viscoelasticity. J. Differ. Eqs.59, 388–412 (1985)Google Scholar
  6. 6.
    Liu, T.-P.: Nonlinear Stability of Shock Waves for Viscous Conservation Laws. Mem. Am. Math. Soc.328, 1–108 (1985)Google Scholar
  7. 7.
    Renardy, M., Hrusa, W., Nohel, J. A.: Mathematical Problems in Viscoelasticity. England: Longman Sci. and Tech., New York: Wiley 1987Google Scholar
  8. 8.
    Dafermos, C. M., Nohel, J. A.: A Nonlinear Hyperbolic volterra Equation in Viscoelasticity. In: Clark, D. N., Pecelli, G., Sacksteder, R. (eds.) Contributions to Analysis and Geometry. Am. J. Math. [Suppl] pp. 87–116. Baltimore, London: The Johns Hopkins University Press 1981Google Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • Yanni Zeng
    • 1
  1. 1.Courant Institute of Mathematical SciencesNew YorkUSA

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