We introduce a notion of viscosity solutions for
a nonlinear degenerate diffusion equation with a drift potential. We
show that our notion of solutions coincide with the weak solutions
defined via integration by parts. As an application of the viscosity
solutions theory, we show that the free boundary uniformly converges
to the equilibrium as $t$ grows. In the case of a convex potential,
an exponential rate of free boundary convergence is obtained.