In this talk I will discuss some results on the long-time dynamics of the solutions to the isentropic Navier-Stokes equations for compressible fluids with a density-dependent diffusion in a bounded interval of the real line.
In particular, I will firstly discuss the stability properties of the (unique) stationary solution by means of the construction of a suitable Lyapunov functional for the system. Subsequently, I will present some results on the asymptotic behavior of the time-dependent solutions, showing how they (slowly) converge towards the aforementioned steady state by firstly developing into a layered function and then by drifting towards it in an exponentially long time interval.

These results are partially contained in [1, 2].