Fritz John pp 532-565 | Cite as

Blow-Up of Solutions of Nonlinear Wave Equations in Three Space Dimensions

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


Let u(x,t) be a solution, □ u ≧ A|u|p for x ∈ ℝ3, t ≧ 0 where □ is the d’Alembertian, and A, p are constants with
$$A > O,1 < p < 1 + \sqrt 2$$
. It is shown that the support of u is contained in the cone 0 ≦ t ≦ tO - |x-xO|, if the “initial data” u(x,O), ut (x,0) have their support in the ball |x-xO| ≦ tO. In particular “global solutions” of u = A|u|p with initial data of compact support vanish identically. On the other hand for
$$A > O,1 > p > 1 + \sqrt 2 $$
global solutions of □ u = A |u|p exist, if the initial data are of compact support and ‖u‖-2 is “sufficiently small” in a suitable norm. For p = 2 the time at which u becomes infinite is of order ‖u‖ -2.


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  1. [1]
    JOHN, F., Partial Differential Equations, 3rd ed., Applied Math. Sciences, Springer-Verlag, New York, 1978Google Scholar
  2. [2]
    BR0WDER, F.E., On non-linear wave equations, Math. Z. 80, 1962, pp. 249–264CrossRefGoogle Scholar
  3. [3]
    GLASSEY, R.T., Blow-up of theorems for nonlinear wave equations, Math. Z. 132, 1973, pp. 183–203CrossRefGoogle Scholar
  4. [4]
    HEINZ, E. and VON WAHL, W., Zu einem Satz von F.E. Browder über nichtlineare Wellengleichungen, Math. Z. 141, 1975, pp. 33–45CrossRefGoogle Scholar
  5. [5]
    JOHN, F., Formation of singularities in one-dimensional nonlinear wave propagation, Comm. Pure Appl. Math. 27, 1974, pp. 377–405CrossRefGoogle Scholar
  6. [6]
    JOHN, F., Delayed singularity formation in solutions of nonlinear wave equations in higher dimensions, Comm. Pure Appl. Math. 29, 1976, pp. 649–682CrossRefGoogle Scholar
  7. [7]
    JÖRGENS, K. Nonlinear wave equations, Lecture Notes, University of Colorado, March 1970Google Scholar
  8. [8]
    JÖRGENS, K., Das Anfangswertproblem in Grossen für eine Klasse nichtlinearer Wellengleichungen, Math. Z. 77, 1961, pp. 295–308CrossRefGoogle Scholar
  9. [9]
    KELLER, J.B., On solutions of nonlinear wave equations, Comm. Pure Appl. Math. 10, 1957, pp. 523–530CrossRefGoogle Scholar
  10. [10]
    KLAINERMAN, S., Global existence for nonlinear wave equations, PreprintGoogle Scholar
  11. [11]
    KNOPS, R.J., LEVINE, H.A. and PAYNE, L.E., Nonexistence, 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. 52–72CrossRefGoogle Scholar
  12. [12]
    LEVINE, H.A., Nonexistence of global weak solutions to some properly and improperly posed problems of mathematical physics; The method of unbounded Fourier coefficients. Math. Ann. 214, 1975, pp. 205–220CrossRefGoogle Scholar
  13. [13]
    LEVINE, H.A., Instability and nonexistence of global solutions to nonlinear wave equations of the form Putt = -Au + F(u). Trans. Amer. Math. Soc. 192, 1974, pp. 1–21Google Scholar
  14. [14]
    LEVINE, H.A., Logarithmic convexity and the Cauchy problem for P(t)utt+ M(t)ut + N(t)u = 0 in Hilbert space. Symposium on non-well-posed problems and logarithmic convexity. Lecture Notes in Math. 316, 1973, pp. 102–160, Springer-VerlagGoogle Scholar
  15. [15]
    LEVINE, H.A. and MURRAY, A., Asymptotic behavior and lower bounds for semilinear wave equations in Hilbert space with applications, SIAM J. Math. Anal.Google Scholar
  16. [16]
    LEVINE, H.A., Logarithmic convexity and the Cauchy problem for some abstract second order differential inequalities. J. Differential Equations 8, 1970, pp. 34–55CrossRefGoogle Scholar
  17. [17]
    LIN, Jeng-Eng and STRAUSS, W., Decay and scattering of solutions of a nonlinear Schrodinger equation, J. Func. Anal. 30, 1978, pp. 245–263CrossRefGoogle Scholar
  18. [18]
    MORAWETZ, C.S., STRAUSS, W.A., Decay and scattering of solutions of a nonlinear relativistic wave equation, Comm. Pure Appl. Math. 25, 1972, pp. 1–31CrossRefGoogle Scholar
  19. [19]
    PAYNE, L.E., Improperly posed problems in partial differential equations, Regional Conference Series in Appl. Math 22, 1975, SIAMGoogle Scholar
  20. [20]
    PAYNE, L.E. and SATTINGER, S.H., Saddle points and instability of nonlinear hyperbolic equations, Israel J. of Math. 22, 1975, pp. 273–303CrossRefGoogle Scholar
  21. [21]
    PECHER, H., Die Existenz regulärer Lösungen für Cauchy- und Anfangs-Randwertprobleme nichtlinearer Wellengleichungen, Math. Z. 140, 1974, pp. 263–279CrossRefGoogle Scholar
  22. [22]
    PECHER, H., Das Verhalten globaler Lösungen nichtlinearer Wellengleichungen für große Zeiten. Math. Z. 136, 1974, pp. 67–92CrossRefGoogle Scholar
  23. [23]
    REED, M., Abstract non-linear wave equations, Lecture Notes in Math., 1976, Springer-VerlagGoogle Scholar
  24. [24]
    SATTINGER, D.H., Stability of nonlinear hyperbolic equations, Arch. Rational Mech Anal. 28, 1968, pp. 226–244CrossRefGoogle Scholar
  25. [25]
    SATTINGER, D.H., On global solutions of nonlinear hyperbolic equations, Arch. Rational Mech. Anal. 30, 1968, pp. 148–172CrossRefGoogle Scholar
  26. [26]
    SEGAL, I.E., Nonlinear semigroups, Ann. of Math. 78, 1963, pp. 339–364CrossRefGoogle Scholar
  27. [27]
    STRAUSS, W.A. , Decay and asymptotics for □ u = F(u), J. Func. Anal. 2, 1968, pp. 409–457CrossRefGoogle Scholar
  28. [28]
    VON WAHL, W., Über die klassische Lösbarkeit des Cauchy-Problems für nichtlineare Wellengleichungen bei kleinen Anfangswerten und das asymtotische Verhalten der Lösungen, Math. Z. 114, 1970, pp. 281–299CrossRefGoogle Scholar
  29. [29]
    VON WAHL, W., Decay estimates for nonlinear wave equations, J. Func. Anal. 9, 1972, pp. 490–495CrossRefGoogle Scholar
  30. [30]
    VON WAHL, W., Ein Anfangswertproblem für hyperbo¬lische Gleichungen mit nichtlinearem elliptischen Hauptteil, Math. Z. 115, 1970, pp. 201–226CrossRefGoogle Scholar
  31. [31]
    VON WAHL, W., Klassische Lösungen nichtlinearer Wellengleichungen im Großen. Math. Z. 112, 1969, pp. 241–279CrossRefGoogle Scholar
  32. [32]
    KATO, T., Blow-up of solutions of some nonlinear hyperbolic equations. PreprintGoogle Scholar
  33. [33]
    STRAUSS, W.A., Oral communicationGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1985

Authors and Affiliations

  • Fritz John
    • 1
  1. 1.Courant Institute of Mathematical SciencesNew YorkUSA

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