Siberian Mathematical Journal

, Volume 59, Issue 6, pp 1034–1050 | Cite as

The Cauchy Problem for the Fractional Diffusion Equation in a Weighted Hölder Space

  • R. M. DzhafarovEmail author
  • N. V. Krasnoshchek


Under study is the Cauchy problem for the fractional diffusion equation with a Caputo derivative. The existence and uniqueness theorems for a smooth solution are proven in a weighted H¨older space.


Caputo derivative fundamental solution Cauchy problem weighted Hölder space 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Diethelm K., The Analysis of Fractional Differential Equations, Springer-Verlag, Berlin (2010).zbMATHGoogle Scholar
  2. 2.
    Hilfer R., Applications of Fractional Analysis in Physics, World Sci., Singapore, New Jersey, London, and Hong Kong (2000).zbMATHGoogle Scholar
  3. 3.
    Tarasov V. E., The Models of Theoretical Physics with Fractional Integro-Differentiation [Russian], Izd. Inst. Kompyuternykh Issledovanii, Moscow and Izhevsk (2011).Google Scholar
  4. 4.
    Langlands T. A. M. and Henry B. I., “Fractional chemotaxis diffusion equation,” Phys. Rev. E., vol. 81, No. 5, 051102 (2010).MathSciNetGoogle Scholar
  5. 5.
    Metzler R. and Klafter J., “The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics,” J. Phys. A., vol. 37, No. 31, 161–208 (2004).MathSciNetzbMATHGoogle Scholar
  6. 6.
    Weiss M., Hashimoto H., and Nilsson T., “Anomalous protein diffusion in living cells as seen by fluorescence correlation spectroscopy,” Biophys. J., vol. 84, No. 6, 4043–4052 (2003).Google Scholar
  7. 7.
    Eidelman S. D. and Kochubei A. N., “Cauchy problem for fractional diffusion equations,” J. Differ. Equ., vol. 199, No. 2, 211–255 (2004).MathSciNetzbMATHGoogle Scholar
  8. 8.
    Kemppainen J. and Ruotsalainen K., “Boundary integral solution of the time-fractional diffusion equation,” Integr. Equ. Oper. Theory, vol. 64, No. 2, 239–249 (2009).MathSciNetzbMATHGoogle Scholar
  9. 9.
    Clément Ph., Londen S.-O., and Simonett G., “Quasilinear evolutionary equations and continuous interpolation spaces,” J. Differ. Equ., vol. 196, No. 2, 418–447 (2004).MathSciNetzbMATHGoogle Scholar
  10. 10.
    Kochubei A. N., “Fractional parabolic systems,” Potential Analysis, vol. 37, No. 1, 1–30 (2012).MathSciNetzbMATHGoogle Scholar
  11. 11.
    Mophou G. M. and N’Guérékata G. M., “On a class of fractional differential equations in a Sobolev space,” Appl. Anal., vol. 91, No. 1, 15–34 (2012).MathSciNetzbMATHGoogle Scholar
  12. 12.
    Pskhu A. V., “The fundamental solution of a diffusion-wave equation of fractional order,” Izv. Math., vol. 73, No. 2, 351–392 (2009).MathSciNetzbMATHGoogle Scholar
  13. 13.
    Zacher R., Quasilinear Parabolic Problems with Nonlinear Boundary Conditions, Ph.D. Thesis, Martin-Luther-Universit ät, Halle and Wittenberg (2003).zbMATHGoogle Scholar
  14. 14.
    Lopushanska H. P., and Lopushanskyj A. O., and Pasichnik E. V., “The Cauchy problem in a space of generalized functions for the equations possessing the fractional time derivative,” Sib. Math. J., vol. 52, No. 6, 1022–1299 (2011).MathSciNetzbMATHGoogle Scholar
  15. 15.
    Sakamoto K. and Yamamoto M., “Initial value boundary value problems for fractional diffusion-wave equations and applications to some inverse problems,” J. Math. Anal. Appl., vol. 382, No. 1, 426–447 (2011).MathSciNetzbMATHGoogle Scholar
  16. 16.
    Allen M., Caffarelli L., and Vasseur A., “A parabolic problem with a fractional time derivative,” Arch. Rat. Mech. Anal., vol. 221, No. 2, 603–630 (2016).MathSciNetzbMATHGoogle Scholar
  17. 17.
    Ponce R., “Hölder continuous solutions for fractional differential equations and maximal regularity,” J. Differ. Equ., vol. 255, No. 10, 3284–3304 (2013).zbMATHGoogle Scholar
  18. 18.
    Belonosov V. S., “Estimates of solutions of parabolic systems in weighted Hölder classes and some of their applications,” Math. USSR-Sb., vol. 38, No. 2, 151–173 (1981).zbMATHGoogle Scholar
  19. 19.
    Solonnikov V. A. and Khachatryan A. G., “Estimates for solutions of parabolic initial-boundary value problems in weighted Hölder norms,” Trudy Mat. Inst. Steklov, vol. 147, 153–162 (1980).zbMATHGoogle Scholar
  20. 20.
    Bizhanova G. I. and Solonnikov V. A., “On the solvability of an initial-boundary value problem for a second-order parabolic equation with a time derivative in the boundary condition in a weighted Hölder space of functions,” St. Petersburg Math. J., vol. 5, No. 1, 97–124 (1994).MathSciNetGoogle Scholar
  21. 21.
    McLean W., “Regularity of solutions to a time-fractional diffusion equation,” ANZIAM J., vol. 52, No. 2, 123–138 (2010).MathSciNetzbMATHGoogle Scholar
  22. 22.
    Bazhlekova E., Jin B., Lazarov R., and Zhou Z., “An analysis of the Rayleigh–Stokes problem for a generalized second- grade fluid,” Numer. Math., vol. 131, No. 1, 1–31 (2015).MathSciNetzbMATHGoogle Scholar
  23. 23.
    Ladyzhenskaya O. A., Solonnikov V. A., and Uraltseva N. N., Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc., Providence (1968).Google Scholar
  24. 24.
    Miranda C., Partial Differential Equations of Elliptic Type [Russian translation], Izdat. Inostr. Lit., Moscow (1957).Google Scholar
  25. 25.
    Kubica A., Rybka R., and Ryszewska K., “Weak solutions of fractional differential equations in a non cylindrical domain,” Nonlinear Anal. Real World Appl., vol. 36, No. 1, 154–182 (2017).MathSciNetzbMATHGoogle Scholar
  26. 26.
    Bazhlekova E., “Subordination principle for fractional evolution equations,” Fractional Calculus Appl. Anal., vol. 3, No. 3, 213–230 (2000).MathSciNetzbMATHGoogle Scholar
  27. 27.
    Pskhu A. V., Fractional Partial Differential Equations [Russian], Nauka, Moscow (2000).Google Scholar
  28. 28.
    Kochubei A. N., “Fractional-order diffusion,” Differ. Uravn., vol. 26, No. 4, 660–670 (1990).zbMATHGoogle Scholar
  29. 29.
    Krasnoschok M. and Vasylyeva N., “On a solvability of a nonlinear fractional reaction-diffusion system in the Hölder spaces,” Nonlinear Stud., vol. 20, No. 4, 589–619 (2013).zbMATHGoogle Scholar
  30. 30.
    Krasnoschok M., “Solvability in Hölder space of an initial-boundary value problem for the time-fractional diffusion equation,” J. Math. Phys., Anal. Geometry, vol. 12, No. 1, 48–77 (2016).MathSciNetzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Institute of Applied Mathematics and MechanicsSlavyanskUkraine

Personalised recommendations