Almost global existence of solutions for the quadratic semilinear Klein-Gordon equation in one space dimension. (English) Zbl 0891.35142

Moriyama, Kazunori; Tonegawa, Satoshi; Tsutsumi, Yoshio

Funkc. Ekvacioj, Ser. Int. 40, No. 2, 313-333 (1997).

The authors consider the lower estimate of the life span of solutions of the Cauchy problem of the Klein-Gordon equation with quadratic nonlinearity in one space dimension when the nonlinearity does not depend on the second derivatives, and the initial conditions depend on a small parameter. The crucial parts in the proof are
(1) to use the normal form technique of Shatah to transform the quadratic nonlinearity in the cubic one, and, thus to establish a priori decay estimates and energy estimates;
(2) to use the decay estimate for the inhomogeneous Klein-Gordon equation due to Georgieu.

Reviewer: L.Vazquez (Madrid)

MSC:

35Q53	KdV equations (Korteweg-de Vries equations)
35L70	Second-order nonlinear hyperbolic equations

Keywords:

Klein-Gordon equation; normal forms; decay estimates; small parameter