Global existence and asymptotics for the quasilinear Klein-Gordon equationwith small data in one space dimension

Let v he a solution to a quasilinear Klein-Gordon equation in one space dim ension squarev + v = F(v, partial derivative (t)v, partial derivative (x)v, partial derivative (t)partial derivative (x)v, partial derivative (2)(x)v) with smooth compactly supported Cauchy data of size epsilon --> 0. Assume that F vanishes at least at order 2 at 0. It is known that the solution v e xists over an interval of time of length larger than e(c/epsilon2) for a po sitive c, and that for a general F it blows up in finite time e(c'/epsilon2 ) (c' > 0). We conjectured in [7] a necessary and sufficient condition on F under which the solution should exist globally in time for small enough ep silon. We prove in this paper the sufficiency of that condition. Moreover, we get a one term asymptotic expansion for u when t --> +infinity. (C) 2001 Editions scientifiques et medicales Elsevier SAS.