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Interpolation Spaces for Initial Values of Abstract Fractional Differential Equations

  • Stig-Olof Londen
Part of the Operator Theory: Advances and Applications book series (OT, volume 168)

Abstract

We consider the parabolic evolutionary equation
$$ D_t^\beta (u_t - z) + Au = f,{\text{ }}u{\text{(0) = }}y,{\text{ }}u_t (0) = z $$
, in continuous interpolation spaces allowing a singularity as t → 0. Here β ∈ (0, 1). We analyze the corresponding trace spaces and show how they depend on β and on the strength of the singularity. The results are applied to the quasilinear equation
$$ D_t^\beta (u_t - z) + A(u,u_t )u = f,{\text{ }}u{\text{(0) = }}y,{\text{ }}u_t (0) = z. $$

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References

  1. [1]
    Ph. Clément, G. Gripenberg, S-O. Londen, Schauder estimates for equations with fractional derivatives, Trans. A.M.S. 352 (2000), 2239–2260.zbMATHCrossRefGoogle Scholar
  2. [2]
    Ph. Clément, G. Gripenberg, S-O. Londen, Regularity properties of solutions of fractional evolution equations. In: “Evolution Equations and their Applications in Physical and Life Sciences”; Proceedings of the Bad Herrenalb Conference, pp. 235–246, Marcel Dekker Lecture Notes in Pure and Applied Mathematics, Vol. 215, 2000.Google Scholar
  3. [3]
    Ph. Clément, S-O. Londen, G. Simonett, Quasilinear evolutionary equations and continuous interpolation spaces, J. Differential Eqs. 196 (2004), 418–447.zbMATHCrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2006

Authors and Affiliations

  • Stig-Olof Londen
    • 1
  1. 1.Institute of MathematicsHelsinki University of TechnologyEspooFinland

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