Loss of hyperbolicity and exponential growth for the viscous Burgers equation with complex forcing terms

We investigate the phenomenon of the time-delay in the instabilities exhibited by the Cauchy problem for the complex viscous Burgers equation on the torus. Precisely, we see that the instantaneous amplification manifested by the solution of the inviscid equation is not observed when introducing a small viscous term in the system. What is more, we show that two distinct phases of the dynamics can be described, that is existence of a bounded solution in times of order one and, after that, an exponential growth in time. This phenomenon is ultimately related to a loss of hyperbolicity and to the subsequent transition to ellipticity for the inviscid problem. The key point of our analysis is a micro-local analysis of the symbol associated to the differential operator and the use of GÃ¥rding's inequality