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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 priori bounds, nodal equilibria and connecting orbits in indefinite superlinear parabolic problems


Authors: Nils Ackermann, Thomas Bartsch, Petr Kaplicky and Pavol Quittner
Journal: Trans. Amer. Math. Soc. 360 (2008), 3493-3539
MSC (2000): Primary 37L05; Secondary 35K20, 35K55, 37L10, 47H20
Published electronically: February 13, 2008
MathSciNet review: 2386234
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Abstract: We consider the dynamics of the semiflow associated with a class of semilinear parabolic problems on a smooth bounded domain, posed with homogeneous Dirichlet boundary conditions. The distinguishing feature of this class is the indefinite superlinear (but subcritical) growth of the nonlinearity at infinity. We present new a priori bounds for global semiorbits that enable us to give dynamical proofs of known and new existence results for equilibria. In addition, we can prove the existence of connecting orbits in many cases.

One advantage of our approach is that the parabolic semiflow is naturally order preserving, in contrast to pseudo-gradient flows considered when using variational methods. Therefore we can obtain much information on nodal properties of equilibria that was not known before.


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Additional Information

Nils Ackermann
Affiliation: Instituto de Matemáticas, Universidad Nacional Autónoma de México, México, D.F. C.P. 04510, México
Email: nils@ackermath.info

Thomas Bartsch
Affiliation: Mathematisches Institut, University of Giessen, Arndtstr. 2, 35392 Giessen, Germany
Email: thomas.bartsch@math.uni-giessen.de

Petr Kaplicky
Affiliation: Faculty of Mathematics and Physics, Charles University Prague, Sokolovská 83, 186 75 Praha 8, Czech Republic
Email: kaplicky@karlin.mff.cuni.cz

Pavol Quittner
Affiliation: Department of Applied Mathematics and Statistics, Comenius University, Mlynská dolina, 84248 Bratislava, Slovakia
Email: quittner@pc2.iam.fmph.uniba.sk

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04404-8
PII: S 0002-9947(08)04404-8
Received by editor(s): July 29, 2004
Received by editor(s) in revised form: April 7, 2006
Published electronically: February 13, 2008
Additional Notes: The first and second authors were supported by DFG Grants BA 1009/15-1, BA 1009/15-2
The third author was supported by the GACR Grant 201/03/0934
The fourth author was supported by the DFG Grant Gi 30/76-1
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.