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

Ackermann, Nils; Bartsch, Thomas; Kaplický, Petr; Quittner, Pavol

doi:10.1090/S0002-9947-08-04404-8

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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
Posted: February 13, 2008
MathSciNet review: 2386234
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Abstract | References | Similar Articles | Additional Information

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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