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The unit condition and global existence for a class of nonlinear Klein- Gordon equations. (English) Zbl 0781.35062
Summary: Global existence for a special class of nonlinear Klein-Gordon equations in two space dimensions is proved. In particular it is shown that, for small initial data, the equation $\partial\sb{tt} u-\Delta u+ u= u\sp 2$ has a global solution.

MSC:
35Q53KdV-like (Korteweg-de Vries) equations
35L15Second order hyperbolic equations, initial value problems
35B65Smoothness and regularity of solutions of PDE
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References:
[1] Christodoulou, D.: Global solutions of nonlinear hyperbolic equations for small initial data. Comm. pure appl. Math. 39, 267-282 (1986) · Zbl 0612.35090
[2] Hörmander, L.: Remarks on the Klein-Gordon equation. Proceedings of the conference on partial differential equations 2 (1987) · Zbl 0655.35057
[3] Klainerman, S.: The null condition and global existence to nonlinear wave equations. Lectures in applied mathematics 23, 293-326 (1986) · Zbl 0599.35105
[4] Klainerman, S.: Global existence of small amplitude solutions to nonlinear Klein-Gordon equations in for space-time dimensions. Comm. pure appl. Math. 38, 631-641 (1985) · Zbl 0597.35100
[5] Klainerman, S.: Uniform decay estimates and Lorentz invariance of the classical wave equation. Comm. pure appl. Math. 38, 321-332 (1985) · Zbl 0635.35059
[6] Kosecki, R.: Long-time existence of the classical solutions to the Klein-Gordon-Dirac equation in three space dimensions. Ph.d. thesis (1987)
[7] Shatah, J.: Normal forms and quadratic nonlinear Klein-Gordon equations. Comm. pure appl. Math. 38, 685-696 (1985) · Zbl 0597.35101