Degenerate diffusion with a drift potential: A viscosity solutionsapproach

Inwon C. Kim; Helen K. Lei

doi:10.3934/dcds.2010.27.767

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2010, Volume 27, Issue 2: 767-786. Doi: 10.3934/dcds.2010.27.767

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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

Received: September 30, 2009

Revised: January 31, 2010

Published: June 2010

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Abstract

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.

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Mathematics Subject Classification: 35D40, 34K21, 35K65.

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