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The Cauchy-Dirichlet problem for the heat equation in Besov spaces

  • E. Zadrzyńska
  • W. M. Zajaczkowski
Article

Abstract

The solvability in anisotropic spaces \( B_{p,q}^{\tfrac{\sigma } {2},\sigma } (\Omega ^T ) \) , σ ∈ ℝ+, p, q ∈ (1, ∞), of the heat equation ut − Δu = f in ΩT ≡ (0, T) × Ω is studied under the boundary and initial conditions u = g on ST, u|t=0 = u0 in Ω, where S is the boundary of a bounded domain Ω ⊂ ℝn. The existence of a unique solution \( B_{p,q}^{\tfrac{\sigma } {2},1,\sigma + 2} (\Omega ^T ) \) of the above problem is proved under the assumptions that \( S \in C^{\sigma + 2} ,f \in B_{p,q}^{\tfrac{\sigma } {2},\sigma } (\Omega ^T ), g \in B_{p,q}^{\tfrac{\sigma } {2} + 1 - \tfrac{1} {{2P}},\sigma + 2 - \tfrac{1} {P}} (S^T ), u_0 \in B_{p,q}^{\sigma + 2 - \tfrac{2} {P}} (\Omega ) \) and under some additional conditions on the data. The existence is proved by the technique of regularizers. For this purpose the local-in-space solvability near the boundary and near an interior point of Ω is needed. To show the local-in-space existence, the definition of Besov spaces by the dyadic decomposition of a partition of unity is used. This enables us to get an appropriate estimate in a new and promising way without applying either the potential technique or the resolvent estimates or the interpolation. Bibliography: 26 titles.

Keywords

Unique Solution Bounded Domain Additional Condition DIRICHLET Problem Interior Point 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media, Inc. 2008

Authors and Affiliations

  1. 1.Faculty of Mathematics and Information SciencesWarsaw University of TechnologyWarsawPoland
  2. 2.Institute of MathematicsPolish Academy of SciencesWarsawPoland

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