  We consider the Cauchy problem for a model of non-linear acoustics, named the
Kuznetsov equation, describing sound propagation in thermo-viscous elastic
media. For the viscous case, it is a weakly quasi-linear strongly damped wave
equation, for which we prove the global existence in time of regular solutions
for sufficiently small initial data, the size of which is specified, and give
the corresponding energy estimates. In the non-viscous case, we update the
known results of John for quasi-linear wave equations, obtaining the
well-posedness results for less regular initial data. We obtain, using a priori
estimates and a Klainerman inequality, the estimations of the maximal existence
time, depending on the space dimension, which are optimal, thanks to the
blow-up results of Alinhac. Alinhac's blow-up results are also confirmed by a L
2-stability estimate, obtained between a regular and a less regular solutions.
