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Lower bound of the time of existence for the nonlinear Klein-Gordon equation in one space dimension. (Minoration du temps d’existence pour l’équation de Klein-Gordon non-linéaire en dimension 1 d’espace.) (French) Zbl 0937.35160

Let \(u\) be a solution to the semilinear Klein-Gordon equation in one space dimension, \[ \partial^2_tu-\partial^2_x u+u=F(u, \partial_tu, \partial_xu, \partial_t \partial_xu, \partial^2_xu) \] where \(F\) is a quadratic nonlinearity and Cauchy data \(u(t=0)=\varepsilon u_0\), \(\partial_t u(t=0)= \varepsilon u\), are small in \(C_0^\infty\). The aim of the paper is to obtain an explicit lower bound of the existence time of the solution \((T)\): \[ \lim\inf_{\varepsilon\to 0}\varepsilon^2\log T_\varepsilon \geq A. \] This is an improvement of the known result for \(A=0\). The expression for \(A\) is directly computed from the Cauchy data and the given nonlinearity. The author first constructs the asymptotic solution and then derives the approximate solution. An interesting constraint on the nonlinearity (called null-condition) is obtained for which \(A\) is \(\infty\).

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35B35 Stability in context of PDEs
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References:

[1] Alinhac, S., Blow-up for nonlinear hyperbolic equations, () · Zbl 1078.35522
[2] Alinhac, S., Explosion des solutions d’une équation d’onde quasi-linéaire en deux dimensions d’espace, Comm. partial diff. eq., 21, 5,6, 923-969, (1996) · Zbl 0858.35082
[3] Alinhac, S., Blow-up of small data solutions for a class of quasilinear wave equations in two space dimensions I, (1996), Université Paris-Sud, preprint
[4] Alinhac, S., Blow-up of small data solutions for a class of quasilinear wave equations in two space dimensions II, (1997), Université Paris-Sud, preprint
[5] Christodoulou, D., Global solutions of nonlinear hyperbolic equations for small initial data, Comm. pure appl. math., 39, 267-282, (1986) · Zbl 0612.35090
[6] Coifman, R.; Meyer, Y., Au-delà des opérateurs pseudo-différentiels, Astérisque, 57, (1978) · Zbl 0483.35082
[7] Georgiev, V., Decay estimates for the Klein-Gordon equations, Comm. partial diff. eq., 17, 1111-1139, (1992) · Zbl 0767.35068
[8] Georgiev, V.; Yordanov, B., Asymptotic behaviour of the one-dimensional Klein-Gordon equation with a cubic nonlinearity, (1997), preprint
[9] Hörmander, L., The lifespan of classical solutions of nonlinear hyperbolic equations, Springer lectures notes in math., 1256, 214-280, (1987)
[10] Hörmander, L., Lectures on nonlinear hyperbolic differential equations, () · Zbl 0881.35001
[11] John, F., Blow-up of radial solutions of utt = c2(ut)δu in three space dimensions, Mat. appl. comput., 4, 3-18, (1985)
[12] John, F.; Klainerman, S., Almost global existence to nonlinear wave equations in three space dimensions, Comm. pure appl. math., 37, 443-455, (1984) · Zbl 0599.35104
[13] Klainerman, S., Global existence for nonlinear wave equations, Comm. pure appl. math., 33, 43-101, (1980) · Zbl 0405.35056
[14] Klainerman, S., Uniform decay estimates and the Lorentz invariance of the classical wave equation, Comm. pure appl. math, 38, 321-332, (1985) · Zbl 0635.35059
[15] Klainerman, S., The null condition and global existence to nonlinear wave equations, Lectures in applied mathematics, 23, 293-326, (1986) · Zbl 0599.35105
[16] Klainerman, S., Global existence of small amplitude solutions to nonlinear Klein-Gordon equations in four space-time dimensions, Comm. pure appl. math., 38, 631-641, (1985) · Zbl 0597.35100
[17] Moriyama, K., Normal forms and global existence of solutions to a class of cubic nonlinear Klein-Gordon equations in one-space dimension, Diff. int. equations, 10, n^{o} 3, 499-520, (1997) · Zbl 0891.35096
[18] Moriyama, K.; Tonegawa, S.; Tsutsumi, Y., Almost global existence of solutions for the quadratic semilinear Klein-Gordon equation in one space dimension, Funkcialaj ekvacioj, 40, n^{o} 2, 313-333, (1997) · Zbl 0891.35142
[19] Ozawa, T.; Tsutaya, K.; Tsutsumi, Y., Global existence and asymptotic behavior of solutions for the Klein-Gordon equations with quadratic nonlinearity in two space dimensions, Math. Z, 222, 341-362, (1996) · Zbl 0877.35030
[20] Shatah, J., Normal forms and quadratic nonlinear Klein-Gordon equations, Comm. pure appl. math., 38, 685-696, (1985) · Zbl 0597.35101
[21] Simon, J.C.H.; Taflin, E., The Cauchy problem for nonlinear Klein-Gordon equations, Commun. math. phys., 152, 433-478, (1993) · Zbl 0783.35066
[22] Yagi, K., Normal forms and nonlinear Klein-Gordon equations in one space dimension, ()
[23] Yordanov, B., Blow-up for the one-dimensional Klein-Gordon equation with a cubic nonlinearity, (1996), preprint
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