# zbMATH — the first resource for mathematics

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 $$\partial_{tt} u-\Delta u+ u= u^ 2$$ has a global solution.

##### MSC:
 35Q53 KdV equations (Korteweg-de Vries equations) 35L15 Initial value problems for second-order hyperbolic equations 35B65 Smoothness and regularity of solutions to PDEs
##### Keywords:
two space dimensions; small initial data
Full Text:
##### 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, (), Saint Jean de Monts · Zbl 0655.35057 [3] Klainerman, S, The null condition and global existence to nonlinear wave equations, (), 293-326 · 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, () [7] Shatah, J, Normal forms and quadratic nonlinear Klein-Gordon equations, Comm. pure appl. math., 38, 685-696, (1985) · Zbl 0597.35101
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.