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May  2010, 27(2): 767-786. doi: 10.3934/dcds.2010.27.767

Degenerate diffusion with a drift potential: A viscosity solutions approach

Inwon C. Kim 1, and  Helen K. Lei 1,

1. 

UCLA Mathematics Department, Box 951555, Los Angeles, CA 90095-1555, United States, United States

Received  October 2009 Revised  February 2010 Published  February 2010

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..
Mathematics Subject Classification: 35D40, 34K21, 35K6.
Citation: Inwon C. Kim, Helen K. Lei. Degenerate diffusion with a drift potential: A viscosity solutions approach. Discrete and Continuous Dynamical Systems, 2010, 27 (2) : 767-786. doi: 10.3934/dcds.2010.27.767
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