Existence of solutions and separation from singularities for a class of fourth order degenerate parabolic equations
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- by Giulio Schimperna and Sergey Zelik PDF
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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.References
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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
- MR Author ID: 357918
- Email: S.Zelik@surrey.ac.uk
- Received by editor(s): September 6, 2010
- Received by editor(s) in revised form: November 15, 2011
- Published electronically: December 13, 2012
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 365 (2013), 3799-3829
- MSC (2010): Primary 35K35, 35K65, 37L30
- DOI: https://doi.org/10.1090/S0002-9947-2012-05824-7
- MathSciNet review: 3042604