A family of nonlinear diffusions connecting Perona-Malik to standard diffusion

Patrick Guidotti

doi:10.3934/dcdss.2012.5.581

Article Contents

2012, Volume 5, Issue 3: 581-590. Doi: 10.3934/dcdss.2012.5.581

This issue Previous Article Some singular perturbations results for semilinear hyperbolic problems Next Article Rate-independent processes with linear growth energies and time-dependent boundary conditions

A family of nonlinear diffusions connecting Perona-Malik to standard diffusion

Patrick Guidotti^1,

1.
Department of Mathematics, University of California, 340 Rowland Hall, Irvine, CA 92697-3975

Received: August 31, 2010

Published: May 2012

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Abstract

A one parameter family of equations is considered which connects the well-known Perona-Malik equation to standard diffusion. The parameter acts as a regularization parameter which gradually modifies the ill-posed Perona-Malik equation, through a strongly locally well-posed equation, to a strongly globally well-posed one which exhibits a behavior akin to that of standard diffusion, which is itself obtained in the limit. In the locally well-posed regime, the equation is degenerate parabolic and the onset of singularities can not be ruled out. Using a classical regularization approach, a-priori estimates can be derived which allow to go to limit with the regularization parameter globally in time. It is, however, not clear how to obtain a proper definition of weak solution for the limiting equation of interest.

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Mathematics Subject Classification: 35A01, 35A02, 35A09, 35K65, 35B40.

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