Degenerate diffusion with a drift potential: a viscosity solutions approach

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 time grows. In the case of a convex potential, an exponential rate of free boundary convergence is obtained.