Fritz John pp 566-588 | Cite as

Blow-Up for Quasi-Linear Wave Equations in Three Space Dimensions

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


We consider equations of the form
$$u = \phi (x,t,u,u',u'')$$
for a function u = u(x 1,…, x n ,t) = u(x,t). Here □ denotes the d’Alembertian
$$\square = \frac{{{\partial ^2}}}{{\partial {t^2}}} - \vartriangle$$
and u′ represents the vector of first derivatives, u″ that of second derivatives of u with respect to the x k and t. On linearizing (1a) shall go over into the classical wave equation □ u = 0; this means that Φ and its first derivatives with respect to u, u′, u″ shall vanish for u = u′ = u″ = 0.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Kato, T., The Cauchy problem for quasi-linear symmetric hyperbolic systems, Archive for Rat. Mech. Analysis, 1974–75, pp. 57–58.Google Scholar
  2. [2]
    Hughes, T. J. R., Kato, T., and Marsden, J. E., Wellposed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity, Archives Rat. Mech. Analysis 63–4, 1976–77, pp. 273–293.Google Scholar
  3. [3]
    John, F., Delayed singularity formation in solutions of nonlinear wave equations in higher dimensions, Comm. Pure Appl. Math. 29, 1976, pp. 649–682.Google Scholar
  4. [4]
    Graff, R. A., A functional analytic approach to existence and uniqueness of solutions to some nonlinear Cauchy problems, preprint.Google Scholar
  5. [5]
    Knops, R. J., Levine, H. A., and Payne, L. E., Non-existence, instability, and growth theorems for solutions of a class of abstract nonlinear equations with applications to nonlinear elastodynamics, Arch. Rational Mech. Anal. 55, 1974, pp. 52–72.CrossRefGoogle Scholar
  6. [6]
    Payne, L. E., Improperly posed problems in partial differential equations, Regional Conference Series in Appl. Math. 22, 1975, SIAM.Google Scholar
  7. [7]
    Pecher, H., Globale klassische Losungen nichtlinearer Wellengleichungen für höhere Raumdimensionen, Nachr. Akad. Wiss. Göttingen Math. Phys., Kl. II, 1975, pp. 221–232.Google Scholar
  8. [8]
    Pecher, H., L p-Abschätzungen und klassische Lösungen für nichtlineare Wellengleichungen. I, Math. Z. 150, 1976, pp. 159–183; II. Manuscripta Math. 20, 1977, pp. 227–244.Google Scholar
  9. [9]
    Pecher, H., Existenzsätze für reguläre Lösungen semilinearer Wellengleichungen, Nachr. Akad. Wiss. Göttingen. Math. Phys. Kl. II, 1979, pp. 129–151.Google Scholar
  10. [10]
    Brenner, P., and von Wahl, W., Global classical solutions of nonlinear wave equations, preprint.Google Scholar
  11. [11]
    Glassey, R. T., Finite-time blow-up for solutions of nonlinear wave equations, preprint.Google Scholar
  12. [12]
    Kato, T., Blow-up of solutions of some nonlinear hyperbolic equations, Comm. Pure Appl. Math. 33, 1980, to appear.Google Scholar
  13. [13]
    John, F., Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math. 28, 1979, pp. 235–268.CrossRefGoogle Scholar
  14. [14]
    Klainerman, S., Global existence for nonlinear wave equations, Comm. Pure Appl. Math. 33, 1980, pp. 43–101.CrossRefGoogle Scholar
  15. [15]
    Klainerman, S., Long time behaviour of solutions to nonlinear evolution equation, preprint.Google Scholar
  16. [16]
    Lax, P. D., Hyperbolic systems of conservation laws and the mathematical theory of shock waves, Regional Conference Series in Applied Mathematics 11, 1973.Google Scholar
  17. [17]
    John, F., Formation of singularities in one-dimensional nonlinear wave propagation, Comm. Pure Appl. Math. 27, 1974, pp. 377–405.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1985

Authors and Affiliations

  • Fritz John

There are no affiliations available

Personalised recommendations