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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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A priori bounds, nodal equilibria and connecting orbits in indefinite superlinear parabolic problems
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by Nils Ackermann, Thomas Bartsch, Petr Kaplický and Pavol Quittner PDF
Trans. Amer. Math. Soc. 360 (2008), 3493-3539 Request permission


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
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  • Thomas Bartsch
  • Affiliation: Mathematisches Institut, University of Giessen, Arndtstr. 2, 35392 Giessen, Germany
  • Email:
  • Petr Kaplický
  • Affiliation: Faculty of Mathematics and Physics, Charles University Prague, Sokolovská 83, 186 75 Praha 8, Czech Republic
  • MR Author ID: 629788
  • Email:
  • Pavol Quittner
  • Affiliation: Department of Applied Mathematics and Statistics, Comenius University, Mlynská dolina, 84248 Bratislava, Slovakia
  • Email:
  • 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
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 360 (2008), 3493-3539
  • MSC (2000): Primary 37L05; Secondary 35K20, 35K55, 37L10, 47H20
  • DOI:
  • MathSciNet review: 2386234