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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
JOHN, F., Partial Differential Equations, 3rd ed., Applied Math. Sciences, Springer-Verlag, New York, 1978
BROWDER, F.E., On non-linear wave equations, Math. Z. 80, 1962, pp. 249–264
GLASSEY, R.T., Blow-up of theorems for nonlinear wave equations, Math. Z. 132, 1973, pp. 183–203
HEINZ, E. and VON WAHL, W., Zu einem Satz von F. E. Browder über nichtlineare Wellengleichungen, Math. Z. 141, 1975, pp. 33–45
JOHN, F., Formation of singularities in one-dimensional nonlinear wave propagation, Comm. Pure Appl. Math. 27, 1974, pp. 377–405
JOHN, F., Delayed singularity formation in solutions of nonlinear wave equations in higher dimensions, Comm. Pure Appl. Math. 29, 1976, pp. 649–682
JÖRGENS, K., Nonlinear wave equations, Lecture Notes, University of Colorado, March 1970
JÖRGENS, K., Das Anfangswertproblem in Grossen für eine Klasse nichtlinearer Wellengleichungen, Math. Z. 77, 1961, pp. 295–308
KELLER, J.B., On solutions of nonlinear wave equations, Comm. Pure Appl. Math. 10, 1957, pp. 523–530
KLAINERMAN, S., Global existence for nonlinear wave equations, Preprint
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
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
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
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
LEVINE, H.A. and MURRAY, A., Asymptotic behavior and lower bounds for semilinear wave equations in Hilbert space with applications, SIAM J. Math. Anal.
LEVINE, H.A., Logarithmic convexity and the Cauchy problem for some abstract second order differential inequalities. J. Differential Equations 8, 1970, pp. 34–55
LIN, Jeng-Eng and STRAUSS, W., Decay and scattering of solutions of a nonlinear Schrodinger equation, J. Func. Anal. 30, 1978, pp. 245–263
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
PAYNE, L.E., Improperly posed problems in partial differential equations, Regional Conference Series in Appl. Math 22, 1975, SIAM
PAYNE, L.E., and SATTINGER, S.H., Saddle points and instability of nonlinear hyperbolic equations, Israel J. of Math. 22, 1975, pp. 273–303
PECHER, H., Die Existenz regulärer Lösungen für Cauchy- und Anfangs-Randwertprobleme nichtlinearer Wellengleichungen, Math. Z. 140, 1974, pp. 263–279
PECHER, H., Das Verhalten globaler Lösungen nichtlinearer Wellengleichungen für große Zeiten. Math. Z. 136, 1974, pp. 67–92
REED, M. Abstract non-linear wave equations, Lecture Notes in Math., 1976, Springer-Verlag
SATTINGER, D.H., Stability of nonlinear hyperbolic equations, Arch. Rational Mech Anal. 28, 1968, pp. 226–244
SATTINGER, D.H., On global solutions of nonlinear hyperbolic equations, Arch. Rational Mech. Anal. 30, 1968, pp. 148–172
SEGAL, I.E., Nonlinear semigroups, Ann. of Math. 78, 1963, pp. 339–364
STRAUSS, W.A., Decay and asymptotics for □u=F(u), J. Func. Anal. 2, 1968, pp. 409–457
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
VON WAHL, W., Decay estimates for nonlinear wave equations, J. Func. Anal. 9, 1972, pp. 490–495
VON WAHL, W., Ein Anfangswertproblem für hyperbolische Gleichungen mit nichtlinearem elliptischen Hauptteil, Math. Z. 115, 1970, pp. 201–226
VON WAHL, W., Klassische Lösungen nichtlinearer Wellengleichungen im Großen. Math. Z. 112, 1969, pp. 241–279
KATO, T., Blow-up of solutions of some nonlinear hyperbolic equations. Preprint
STRAUSS, W.A., Oral communication
Author information
Authors and Affiliations
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
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
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01647974