Almost global existence for solution of semilinear klein-gordon equations with small weakly decaying cauchy data

Abstract

Let T∈ be the time of exisstence of a semilinear Klein-Gordon equation with small,smooth,Cauchy data of size ∈ in space dimension d≥2 If the Cauchy data are decaying rapidly enough at infinity,and the nonlinearity vanishes at least at order 2 at 0,it is well known that T∈=+∞ for∈ small enough. The aim of this paper is to show that if one assumes only a weak decay of the Cauchy data at infinity,one has a lower bound T∈≤Cexp(c∈−μ)(μ=2/3 if d =2,μ=1 id d≤3) when the nonlinearity satisfies a convenient "null condition"