In this talk I will discuss the influence of a regularizing term in the instabilities of some hyperbolic systems; in particular, I will show that, for a class of hyperbolic systems that are strongly unstable (i.e., for which an instability is manifested as soon as t>0), a small regularizing term introduces a time-delay in the aforementioned instability.

This behavior is ultimately related to a change in the behavior of the equation under consideration, which presents a transition from hyperbolicity to ellipticity for t> 0.

The main example I will show is a viscous regularization of the Burgers equation with complex forcing terms. Finally, I will mention what happens when regularizing the Euler equations with a dispersive term.

Some of these results have been obtained in collaboration with B. Texier.