# American Institute of Mathematical Sciences

June  2012, 5(3): 581-590. doi: 10.3934/dcdss.2012.5.581

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

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

Received  September 2010 Published  October 2011

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.
Citation: Patrick Guidotti. A family of nonlinear diffusions connecting Perona-Malik to standard diffusion. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 581-590. doi: 10.3934/dcdss.2012.5.581
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##### References:
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