In this paper we deal with the long time existence for the Cauchy problem
associated to some asymptotic models for long wave, small amplitude gravity
surface waves. We generalize some of the results that can be found in the
literature devoted to the study of Boussinesq systems by implementing an energy
method on spectrally localized equations. In particular, we obtain better
results in terms of the regularity level required to solve the initial value
problem on large time scales