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Blow-up of solutions of nonlinear wave equations in three space dimensions. (English) Zbl 0406.35042

MSC:
35L30 Initial value problems for higher-order hyperbolic equations
35L70 Second-order nonlinear hyperbolic equations
35A30 Geometric theory, characteristics, transformations in context of PDEs
35L05 Wave equation
35L60 First-order nonlinear hyperbolic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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References:
[1] JOHN, F., Partial Differential Equations, 3rd ed., Applied Math. Sciences, Springer-Verlag, New York, 1978 · Zbl 0426.35002
[2] BROWDER, F.E., On non-linear wave equations, Math. Z. 80, 1962, pp. 249–264 · Zbl 0109.32102 · doi:10.1007/BF01162382
[3] GLASSEY, R.T., Blow-up of theorems for nonlinear wave equations, Math. Z. 132, 1973, pp. 183–203 · Zbl 0254.35078 · doi:10.1007/BF01213863
[4] HEINZ, E. and VON WAHL, W., Zu einem Satz von F. E. Browder über nichtlineare Wellengleichungen, Math. Z. 141, 1975, pp. 33–45 · Zbl 0289.35076 · doi:10.1007/BF01236982
[5] JOHN, F., Formation of singularities in one-dimensional nonlinear wave propagation, Comm. Pure Appl. Math. 27, 1974, pp. 377–405 · Zbl 0302.35064 · doi:10.1002/cpa.3160270307
[6] JOHN, F., Delayed singularity formation in solutions of nonlinear wave equations in higher dimensions, Comm. Pure Appl. Math. 29, 1976, pp. 649–682 · Zbl 0332.35044 · doi:10.1002/cpa.3160290608
[7] JÖRGENS, K., Nonlinear wave equations, Lecture Notes, University of Colorado, March 1970
[8] JÖRGENS, K., Das Anfangswertproblem in Grossen für eine Klasse nichtlinearer Wellengleichungen, Math. Z. 77, 1961, pp. 295–308 · Zbl 0111.09105 · doi:10.1007/BF01180181
[9] KELLER, J.B., On solutions of nonlinear wave equations, Comm. Pure Appl. Math. 10, 1957, pp. 523–530 · Zbl 0090.31802 · doi:10.1002/cpa.3160100404
[10] KLAINERMAN, S., Global existence for nonlinear wave equations, Preprint · Zbl 0405.35056
[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 · Zbl 0292.35067 · doi:10.1007/BF00282433
[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 · Zbl 0293.35003 · doi:10.1007/BF01352106
[13] LEVINE, H.A., Instability and nonexistence of global solutions to nonlinear wave equations of the form Putt=u+F(u). Trans. Amer. Math. Soc. 192, 1974, pp. 1–21 · Zbl 0288.35003
[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 · doi:10.1007/BFb0069626
[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. · Zbl 0308.35010
[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 · Zbl 0194.13101 · doi:10.1016/0022-0396(70)90038-0
[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 · Zbl 0395.35070 · doi:10.1016/0022-1236(78)90073-3
[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 · Zbl 0228.35055 · doi:10.1002/cpa.3160250103
[19] PAYNE, L.E., Improperly posed problems in partial differential equations, Regional Conference Series in Appl. Math 22, 1975, SIAM
[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 · Zbl 0317.35059 · doi:10.1007/BF02761595
[21] PECHER, H., Die Existenz regulärer Lösungen für Cauchy- und Anfangs-Randwertprobleme nichtlinearer Wellengleichungen, Math. Z. 140, 1974, pp. 263–279 · Zbl 0287.35069 · doi:10.1007/BF01214167
[22] PECHER, H., Das Verhalten globaler Lösungen nichtlinearer Wellengleichungen für große Zeiten. Math. Z. 136, 1974, pp. 67–92 · Zbl 0269.35059 · doi:10.1007/BF01189258
[23] REED, M. Abstract non-linear wave equations, Lecture Notes in Math., 1976, Springer-Verlag · Zbl 0317.35002
[24] SATTINGER, D.H., Stability of nonlinear hyperbolic equations, Arch. Rational Mech Anal. 28, 1968, pp. 226–244 · Zbl 0157.17201 · doi:10.1007/BF00250928
[25] SATTINGER, D.H., On global solutions of nonlinear hyperbolic equations, Arch. Rational Mech. Anal. 30, 1968, pp. 148–172 · Zbl 0159.39102 · doi:10.1007/BF00250942
[26] SEGAL, I.E., Nonlinear semigroups, Ann. of Math. 78, 1963, pp. 339–364 · Zbl 0204.16004 · doi:10.2307/1970347
[27] STRAUSS, W.A., Decay and asymptotics for =F(u), J. Func. Anal. 2, 1968, pp. 409–457 · Zbl 0182.13602 · doi:10.1016/0022-1236(68)90004-9
[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 · Zbl 0186.17001 · doi:10.1007/BF01112698
[29] VON WAHL, W., Decay estimates for nonlinear wave equations, J. Func. Anal. 9, 1972, pp. 490–495 · Zbl 0229.35054 · doi:10.1016/0022-1236(72)90023-7
[30] VON WAHL, W., Ein Anfangswertproblem für hyperbolische Gleichungen mit nichtlinearem elliptischen Hauptteil, Math. Z. 115, 1970, pp. 201–226 · Zbl 0191.11402 · doi:10.1007/BF01109859
[31] VON WAHL, W., Klassische Lösungen nichtlinearer Wellengleichungen im Großen. Math. Z. 112, 1969, pp. 241–279 · Zbl 0177.36602 · doi:10.1007/BF01110242
[32] KATO, T., Blow-up of solutions of some nonlinear hyperbolic equations. Preprint · Zbl 0421.35053
[33] STRAUSS, W.A., Oral communication
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