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

 

Existence of solutions and separation from singularities for a class of fourth order degenerate parabolic equations


Authors: Giulio Schimperna and Sergey Zelik
Journal: Trans. Amer. Math. Soc. 365 (2013), 3799-3829
MSC (2010): Primary 35K35, 35K65, 37L30
Published electronically: December 13, 2012
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Abstract: A nonlinear parabolic equation of the fourth order is analyzed. The equation is characterized by a mobility coefficient that degenerates at 0. Existence of at least one weak solution is proved by using a regularization procedure and deducing suitable a priori estimates. If a viscosity term is added and additional conditions on the nonlinear terms are assumed, then it is proved that any weak solution becomes instantaneously strictly positive. This in particular implies uniqueness for strictly positive times and further time-regularization properties. The long-time behavior of the problem is also investigated and the existence of trajectory attractors and, under more restrictive conditions, of strong global attractors is shown.


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Additional Information

Giulio Schimperna
Affiliation: Dipartimento di Matematica, Università di Pavia, Via Ferrata 1, I-27100 Pavia, Italy
Email: giusch04@unipv.it

Sergey Zelik
Affiliation: Department of Mathematics, University of Surrey, Guildford, GU2 7XH, United Kingdom
Email: S.Zelik@surrey.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05824-7
PII: S 0002-9947(2012)05824-7
Received by editor(s): September 6, 2010
Received by editor(s) in revised form: November 15, 2011
Published electronically: December 13, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.