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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A class of local classical solutions for the one-dimensional Perona-Malik equation


Authors: Marina Ghisi and Massimo Gobbino
Journal: Trans. Amer. Math. Soc. 361 (2009), 6429-6446
MSC (2000): Primary 35A07, 35B65, 35K65
Published electronically: June 17, 2009
MathSciNet review: 2538599
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Abstract: We consider the Cauchy problem for the one-dimensional Perona-Malik equation

$\displaystyle u_{t}=\frac{1-u_{x}^{2}}{(1+u_{x}^{2})^{2}} u_{xx}$

in the interval $ [-1,1]$, with homogeneous Neumann boundary conditions.

We prove that the set of initial data for which this equation has a local-in-time classical solution $ u:[-1,1]\times[0,T]\to\mathbb{R}$ is dense in $ C^{1}([-1,1])$. Here ``classical solution'' means that $ u$, $ u_{t}$, $ u_{x}$ and $ u_{xx}$ are continuous functions in $ [-1,1]\times[0,T]$.


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Additional Information

Marina Ghisi
Affiliation: Dipartimento di Matematica “Leonida Tonelli”, Università degli Studi di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy
Email: ghisi@dm.unipi.it

Massimo Gobbino
Affiliation: Dipartimento di Matematica Applicata “Ulisse Dini”, Università degli Studi di Pisa, Via Filippo Buonarroti 1c, 56127 Pisa, Italy
Email: m.gobbino@dma.unipi.it

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04793-X
PII: S 0002-9947(09)04793-X
Keywords: Perona-Malik equation, classical solution, forward-backward parabolic equation, anisotropic diffusion, supersolutions, comparison principles.
Received by editor(s): November 13, 2006
Received by editor(s) in revised form: October 25, 2007
Published electronically: June 17, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.