Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Scale-invariant extinction time estimates for some singular diffusion equations

Pages: 509 - 535, Volume 30, Issue 2, June 2011      doi:10.3934/dcds.2011.30.509

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Yoshikazu Giga - Graduate School of Mathematical Sciences, University of Tokyo, Komaba 3-8-1, Tokyo 153-8914, Japan (email)
Robert V. Kohn - Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, United States (email)

Abstract: We study three singular parabolic evolutions: the second-order total variation flow, the fourth-order total variation flow, and a fourth-order surface diffusion law. Each has the property that the solution becomes identically zero in finite time. We prove scale-invariant estimates for the extinction time, using a simple argument which combines an energy estimate with a suitable Sobolev-type inequality.

Keywords:  Total variation flow, extinction time, surface diffusion.
Mathematics Subject Classification:  Primary: 35K15, 35K30; Secondary: 35K55, 35B40.

Received: July 2010;      Revised: July 2010;      Available Online: February 2011.