Nonlinear Cauchy problem with slowly oscillating initial data

We solve the Cauchy problem with weakly oscillating data, in the space of h olomorphic functions with respect to time, and Gevrey class with respect to the other variables. We show the stability of Cauchy problem's solution wi th nul initial data. In the stationary case, we give estimates of the solut ion's lifespan with respect to the speed's size of the oscillations of init ial data. (C) 2001 Editions scientifiques et medicales Elsevier SAS.