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GEM - International Journal on Geomathematics

, Volume 9, Issue 2, pp 293–315 | Cite as

Describing the singular behaviour of parabolic equations on cones in fractional Sobolev spaces

  • Stephan Dahlke
  • Cornelia Schneider
Original Paper
  • 22 Downloads

Abstract

In this paper, the Dirichlet problem for parabolic equations in a wedge is considered. In particular, we study the smoothness of the solutions in the fractional Sobolev scale \(H^s\), \(s\in \mathbb {R}\). The regularity in these spaces is related with the approximation order that can be achieved by numerical schemes based on uniform grid refinements. Our results provide a first attempt to generalize the well-known \(H^{3/2}\)-Theorem of Jerison and Kenig (J Funct Anal 130:161–219, 1995) to parabolic PDEs. As a special case the heat equation on radial-symmetric cones is investigated.

Keywords

Parabolic evolution equations Regularity theory Fractional Sobolev spaces 

Mathematics Subject Classification

Primary: 35B65 35K10 Secondary: 46E35 

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.FB12 Mathematik und InformatikPhilipps-Universität MarburgMarburgGermany
  2. 2.Department Mathematik, AM3Friedrich-Alexander-Universität Erlangen-NürnbergErlangenGermany

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