Almost global existence for solutions of semilinear Klein-Gordon equationswith small weakly decaying Cauchy data.

Infinity, and the nonlinearity vanishes at least at order 2 at 0, it is wel l known that T epsilon = +infinity for a small enough. The aim of this gape r is to show that if one assumes only a weak decay of the Cauchy data at in finity, one has a lower bound T-epsilon greater than or equal to cexp(c eps ilon (-u)) (mu = 2/3 if d = 2, mu = 1 if d greater than or equal to 3) when the nonlinearity satisfies a convenient "null condition".