What is the characteristic time of the evaporation of the galaxies?
The standard treatment can be found in (Binney &amp; Tremaine 2008), but see also (Adams &amp; Laughlin 1997) for a good treatment. 

The overall timescale for galactic evaporation is $$\tau_{evap}= 100\tau_{relax}\sim 10^{19}$$ years.

The relaxation timescale $$\tau_{relax}=\frac{R}{v}\frac{N}{12 \ln(N/2)},$$ where $R$ is the size of the system, $v$ is the typical random velocity, and $N$ is the total number of stars. This corresponds to the time it takes to completely randomize the velocity of a star by interactions with other stars.