Degenerate diffusion with a drift potential: A viscosity solutions approach
Inwon C. Kim Helen K. Lei
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.
keywords: well-posedness porous medium equation degenerate diffusion viscosity solutions convergence to equilibrium.

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