  We prove the large time behavior of solutions to a coupled thermo-diffusion
arising in the modelling of the motion of hot colloidal particles in porous
media. Additionally, we also ensure the uniqueness of solutions of the target
problem. The main mathematical difficulty is due to the presence in the
right-hand side of the equations of products between temperature and
concentration gradients. Such terms mimic the so-called thermodynamic Soret and
Dufour effects. These are cross-coupling terms emphasizing in this context a
strong interplay between heat conduction and molecular diffusion.
