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

  • E. Zadrzyńska
  • W. M. Zajaczkowski


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.


Unique Solution Bounded Domain Additional Condition DIRICHLET Problem Interior Point 
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  1. 1.
    M. S. Agranovich and M. I. Vishik, “Elliptic problems with parameter and parabolic problems of general type,” Usp. Mat. Nauk, 19, 3 (117), 53–161 (1964).MATHGoogle Scholar
  2. 2.
    W. Alame, “On the existence of solutions for the nonstationary Stokes system with slip boundary conditions in general Sobolev-Slobodetskii and Besov spaces, regularity and other aspects of the Navier-Stokes equations, ” Banach Center Publ., 70, 21–49 (2005).MathSciNetCrossRefGoogle Scholar
  3. 3.
    H. Amann, “Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems,” Diff. Int. Eqs., 3(1), 13–75 (1990).MATHMathSciNetGoogle Scholar
  4. 4.
    H. Amann, Linear and Quasilinear Parabolic Problems, Vol. I, Birkhäuser Verlag (1995).Google Scholar
  5. 5.
    H. Amann, “Elliptic operators with infinite-dimensional state spaces,” J. Evol. Eq., 1, 143–188 (2001).MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    O. V. Besov, V. P. Il’in, and S. M. Nikolskij, Integral Representation of Functions and Imbedding Theorems [in Russian], Nauka, Moscow (1975).Google Scholar
  7. 7.
    M. Burnat and W. M. Zajaczkowski, “On local motion of a compressible barotropic viscous fluid with the boundary slip condition,” Topol. Meth. Nonlin. Anal., 10, 195–223 (1997).MATHMathSciNetGoogle Scholar
  8. 8.
    R. Danchin, “Global existence in critical spaces for flows of compressible viscous and heat-conductive gases,” Arch. Rat. Mech. Anal., 160, 1–39 (2001).MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    R. Danchin, “On the uniqueness in critical spaces for compressible Navier-Stokes equations,” No. DEA, 12, 111–128 (2005).MATHMathSciNetGoogle Scholar
  10. 10.
    G. Grubb, “Functional calculus of pseudodifferential boundary problems,” Progr. Math., 65, Birkhauser (1996).Google Scholar
  11. 11.
    G. Grubb and V. A. Solonnikov, “Solution of parabolic pseudo-differential initial-boundary value problems,” J. Diff. Eqs., 87, 256–304 (1990).MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    G. Grubb and N. J. Kokholm, “A global calculus of parameter-dependent pseudodifferential boundary problems in L p Sobolev spaces,” Acta Math., 171, 165–229 (1993).MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    G. Grubb, “Parameter-elliptic and parabolic pseudodifferential boundary problems in global L p Sobolev spaces,” Math. Zeitschrift, 218, 43–90 (1995).MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    G. Grubb and V. A. Solonnikov, “Boundary value problems for the nonstationary Navier-Stokes equations treated by pseudo-differential methods,” Math. Scand., 69, 217–290 (1991).MATHMathSciNetGoogle Scholar
  15. 15.
    O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type [in Russian], Nauka, Moscow (1967).MATHGoogle Scholar
  16. 16.
    J. L. Lions and E. Magenes, Nonhomogeneous Boundary-Value Problems and Applications, Vol. 1 and 2, Springer Verlag (1972).Google Scholar
  17. 17.
    S. M. Nikolskij, Approximation of Functions with many Variables and Imbedding Theorems [in Russian], Nauka, Moscow (1977).Google Scholar
  18. 18.
    R. Paley and N. Wiener, Fourier Transforms in the Complex Domain, New York (1934).Google Scholar
  19. 19.
    V. A. Solonnikov, “A priori estimates for linear parabolic equations of the second order,” Tr. Steklov Mat. Inst., 70, 133–212 (1964).MATHMathSciNetGoogle Scholar
  20. 20.
    V. A. Solonnikov, “On boundary value problems for linear parabolic systems of differential equations of general type,” Tr. Steklov Mat. Inst., 83 (1965).Google Scholar
  21. 21.
    V. A. Solonnikov, “An initial boundary-value problem for a Stokes system that arises in the study of free boundary problem,” Tr. Steklov Mat. Inst., 188, 150–188 (1990).MathSciNetGoogle Scholar
  22. 22.
    H. Triebel, Interpolation Theory, Function spaces, and Differential Operators, North Holland, Amsterdam (1978).Google Scholar
  23. 23.
    H. Triebel, Theory of Function Spaces, Akad. Verlagsgesellschaft, Leipzig (1983).Google Scholar
  24. 24.
    P. Weidemaier, “Maximal regularity for parabolic equations with inhomogeneous boundary conditions in Sobolev spaces with mixed L p-norm,” Electr. Res. Announ. Amer. Math. Soc., 8, 47–51 (2002).MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    P. Weidemaier, “Existence results in L pL q spaces for second order parabolic equations with inhomogeneous Dirichlet boundary conditions,” Pitman Research Notes, 384, 189–200 (1998).MathSciNetGoogle Scholar
  26. 26.
    Y. Yamamoto, “Solutions in Besov spaces of a class of abstract parabolic equations of higher order in time,” J. Math. Kyoto Univ., 38, 2, 201–227 (1998).MATHMathSciNetGoogle Scholar

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© 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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