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Fritz John pp 624-641 | Cite as

Formation of Singularities in Elastic Waves

  • Fritz John
Chapter
Part of the Contemporary Mathematicians book series (CM)

Abstract

This paper deals with the radial solutions of the dynamic equations for an isotropic homogeneous hyperelastic medium. It is shown that nontrivial solutions “blow up” (cease to exist in the proper sense) after a finite time, if:
  1. (a)

    The equations satisfy a certain “genuine nonlinearity condition”.

     
  2. (b)

    The initial data have compact support and are “sufficiently small”.

     

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References

  1. [1]
    John, F., “Formation of singularities in one-dimensional nonlinear wave propagation,” Comm. Pure Appl. Math. 27 (1974), 377–405.CrossRefGoogle Scholar
  2. [2]
    Klainerman, S., “On ‘almost global’ solutions to quasilinear wave equations in three space dimensions,” Comm. Pure Appl. Math. 36 (1983), 325–344.CrossRefGoogle Scholar
  3. [3]
    Ball, J. M., “Differentiability properties of symmetric and isotropic functions.”Google Scholar
  4. [4]
    John, F., “Finite amplitude waves in a homogeneous isotropic elastic solid,” Comm. Pure Appl. Math. 30 (1977), 421–446.CrossRefGoogle Scholar
  5. [5]
    John, F., “Instability of finite amplitude elastic waves,” Proc. IUTAM Sym. on Finite Elasticity, Martinus Nijhoff, 1981.Google Scholar
  6. [6]
    John, F., “Blow-up for quasi-linear wave equations in three space dimensions,” Comm. Pure Appl. Math. 34 (1981), 29–51.CrossRefGoogle Scholar
  7. [7]
    Sideris, T. C., “Global behavior of solutions of nonlinear equations in three dimensions”, Comm. P. D. E. 8 (1983), 1291–1323.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1985

Authors and Affiliations

  • Fritz John

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