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, Volume 28, Issue 1–3, pp 235–268 | Cite as

Blow-up of solutions of nonlinear wave equations in three space dimensions

  • Fritz John
Article

Abstract

Let u(x,t) be a solution, □ u≧A|u|p for x∈IR3, t≧0 where □ is the d'Alembertian, and A, p are constants with A>0, 1<p<1+√2. It is shown that the support of u is contained in the cone 0≦t≦t0−|x−x0|, if the “initial data” u(x,0), ut(x,0) have their support in the ball |x−x0|≦t0. In particular “global solutions” of u=A|u|p with initial data of compact support vanish identically. On the other hand for A>0, p>1+√2 global solutions of □u=A|u|p exist, if the initial data are of compact support and ∥u∥ is “sufficiently small” in a suitable norm. For p=2 the time at which u becomes infinite is of order ∥u∥−2.

Keywords

Initial Data Wave Equation Number Theory Compact Support Algebraic Geometry 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    JOHN, F., Partial Differential Equations, 3rd ed., Applied Math. Sciences, Springer-Verlag, New York, 1978Google Scholar
  2. [2]
    BROWDER, F.E., On non-linear wave equations, Math. Z. 80, 1962, pp. 249–264Google Scholar
  3. [3]
    GLASSEY, R.T., Blow-up of theorems for nonlinear wave equations, Math. Z. 132, 1973, pp. 183–203Google Scholar
  4. [4]
    HEINZ, E. and VON WAHL, W., Zu einem Satz von F. E. Browder über nichtlineare Wellengleichungen, Math. Z. 141, 1975, pp. 33–45Google Scholar
  5. [5]
    JOHN, F., Formation of singularities in one-dimensional nonlinear wave propagation, Comm. Pure Appl. Math. 27, 1974, pp. 377–405Google Scholar
  6. [6]
    JOHN, F., Delayed singularity formation in solutions of nonlinear wave equations in higher dimensions, Comm. Pure Appl. Math. 29, 1976, pp. 649–682Google 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. 295–308Google Scholar
  9. [9]
    KELLER, J.B., On solutions of nonlinear wave equations, Comm. Pure Appl. Math. 10, 1957, pp. 523–530Google 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. 52–72Google 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. 205–220Google 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. 34–55Google 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. 245–263Google 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. 1–31Google 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. 273–303Google Scholar
  21. [21]
    PECHER, H., Die Existenz regulärer Lösungen für Cauchy- und Anfangs-Randwertprobleme nichtlinearer Wellengleichungen, Math. Z. 140, 1974, pp. 263–279Google Scholar
  22. [22]
    PECHER, H., Das Verhalten globaler Lösungen nichtlinearer Wellengleichungen für große Zeiten. Math. Z. 136, 1974, pp. 67–92Google 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. 226–244Google Scholar
  25. [25]
    SATTINGER, D.H., On global solutions of nonlinear hyperbolic equations, Arch. Rational Mech. Anal. 30, 1968, pp. 148–172Google Scholar
  26. [26]
    SEGAL, I.E., Nonlinear semigroups, Ann. of Math. 78, 1963, pp. 339–364Google Scholar
  27. [27]
    STRAUSS, W.A., Decay and asymptotics for □u=F(u), J. Func. Anal. 2, 1968, pp. 409–457Google 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. 281–299Google Scholar
  29. [29]
    VON WAHL, W., Decay estimates for nonlinear wave equations, J. Func. Anal. 9, 1972, pp. 490–495Google Scholar
  30. [30]
    VON WAHL, W., Ein Anfangswertproblem für hyperbolische Gleichungen mit nichtlinearem elliptischen Hauptteil, Math. Z. 115, 1970, pp. 201–226Google Scholar
  31. [31]
    VON WAHL, W., Klassische Lösungen nichtlinearer Wellengleichungen im Großen. Math. Z. 112, 1969, pp. 241–279Google 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-Verlag 1979

Authors and Affiliations

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

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