Advertisement

Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

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

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.

This is a preview of subscription content, log in to check access.

References

  1. [1]

    JOHN, F., Partial Differential Equations, 3rd ed., Applied Math. Sciences, Springer-Verlag, New York, 1978

  2. [2]

    BROWDER, F.E., On non-linear wave equations, Math. Z. 80, 1962, pp. 249–264

  3. [3]

    GLASSEY, R.T., Blow-up of theorems for nonlinear wave equations, Math. Z. 132, 1973, pp. 183–203

  4. [4]

    HEINZ, E. and VON WAHL, W., Zu einem Satz von F. E. Browder über nichtlineare Wellengleichungen, Math. Z. 141, 1975, pp. 33–45

  5. [5]

    JOHN, F., Formation of singularities in one-dimensional nonlinear wave propagation, Comm. Pure Appl. Math. 27, 1974, pp. 377–405

  6. [6]

    JOHN, F., Delayed singularity formation in solutions of nonlinear wave equations in higher dimensions, Comm. Pure Appl. Math. 29, 1976, pp. 649–682

  7. [7]

    JÖRGENS, K., Nonlinear wave equations, Lecture Notes, University of Colorado, March 1970

  8. [8]

    JÖRGENS, K., Das Anfangswertproblem in Grossen für eine Klasse nichtlinearer Wellengleichungen, Math. Z. 77, 1961, pp. 295–308

  9. [9]

    KELLER, J.B., On solutions of nonlinear wave equations, Comm. Pure Appl. Math. 10, 1957, pp. 523–530

  10. [10]

    KLAINERMAN, S., Global existence for nonlinear wave equations, Preprint

  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. 52–72

  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. 205–220

  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. 1–21

  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-Verlag

  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.

  16. [16]

    LEVINE, H.A., Logarithmic convexity and the Cauchy problem for some abstract second order differential inequalities. J. Differential Equations 8, 1970, pp. 34–55

  17. [17]

    LIN, Jeng-Eng and STRAUSS, W., Decay and scattering of solutions of a nonlinear Schrodinger equation, J. Func. Anal. 30, 1978, pp. 245–263

  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. 1–31

  19. [19]

    PAYNE, L.E., Improperly posed problems in partial differential equations, Regional Conference Series in Appl. Math 22, 1975, SIAM

  20. [20]

    PAYNE, L.E., and SATTINGER, S.H., Saddle points and instability of nonlinear hyperbolic equations, Israel J. of Math. 22, 1975, pp. 273–303

  21. [21]

    PECHER, H., Die Existenz regulärer Lösungen für Cauchy- und Anfangs-Randwertprobleme nichtlinearer Wellengleichungen, Math. Z. 140, 1974, pp. 263–279

  22. [22]

    PECHER, H., Das Verhalten globaler Lösungen nichtlinearer Wellengleichungen für große Zeiten. Math. Z. 136, 1974, pp. 67–92

  23. [23]

    REED, M. Abstract non-linear wave equations, Lecture Notes in Math., 1976, Springer-Verlag

  24. [24]

    SATTINGER, D.H., Stability of nonlinear hyperbolic equations, Arch. Rational Mech Anal. 28, 1968, pp. 226–244

  25. [25]

    SATTINGER, D.H., On global solutions of nonlinear hyperbolic equations, Arch. Rational Mech. Anal. 30, 1968, pp. 148–172

  26. [26]

    SEGAL, I.E., Nonlinear semigroups, Ann. of Math. 78, 1963, pp. 339–364

  27. [27]

    STRAUSS, W.A., Decay and asymptotics for □u=F(u), J. Func. Anal. 2, 1968, pp. 409–457

  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. 281–299

  29. [29]

    VON WAHL, W., Decay estimates for nonlinear wave equations, J. Func. Anal. 9, 1972, pp. 490–495

  30. [30]

    VON WAHL, W., Ein Anfangswertproblem für hyperbolische Gleichungen mit nichtlinearem elliptischen Hauptteil, Math. Z. 115, 1970, pp. 201–226

  31. [31]

    VON WAHL, W., Klassische Lösungen nichtlinearer Wellengleichungen im Großen. Math. Z. 112, 1969, pp. 241–279

  32. [32]

    KATO, T., Blow-up of solutions of some nonlinear hyperbolic equations. Preprint

  33. [33]

    STRAUSS, W.A., Oral communication

Download references

Author information

Additional information

Dedicated to Hans Lewy and Charles B. Morrey, Jr.

The research for this paper was performed at the Courant Institute and supported by the Office of Naval Research under Contract No. N00014-76-C-0301. Reproduction in whole or part is permitted for any purpose of the United States Government.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

John, F. Blow-up of solutions of nonlinear wave equations in three space dimensions. Manuscripta Math 28, 235–268 (1979). https://doi.org/10.1007/BF01647974

Download citation

Keywords

  • Initial Data
  • Wave Equation
  • Number Theory
  • Compact Support
  • Algebraic Geometry