On existence and uniqueness of viscosity solutions for second order fully nonlinear PDEs with Caputo time fractional derivatives

Article
  • 39 Downloads

Abstract

Initial-boundary value problems for second order fully nonlinear PDEs with Caputo time fractional derivatives of order less than one are considered in the framework of viscosity solution theory. Associated boundary conditions are Dirichlet and Neumann, and they are considered in the strong sense and the viscosity sense, respectively. By a comparison principle and Perron’s method, unique existence for the Cauchy–Dirichlet and Cauchy–Neumann problems are proved.

Keywords

Caputo time fractional derivatives Initial-boundary value problems Second order fully nonlinear equations Viscosity solutions 

Mathematics Subject Classification

35D40 35K20 35R11 

Notes

Acknowledgements

Part of the result in this paper was established while the author was visiting the University of Warsaw during May and June 2017. Its hospitality is gratefully acknowledged. The author would like to thank Dr. Yikan Liu for letting him know references on linear equations with Caputo time fractional derivatives. This work is supported by Grant-in-aid for Scientific Research of JSPS Fellows No. 16J03422 and partly the Program for Leading Graduate Schools, MEXT, Japan.

References

  1. 1.
    Allen, M.: A nondivergence parabolic problem with a fractional time derivative. preprint. arXiv:1507.04324 [math.AP]
  2. 2.
    Allen, M., Caffarelli, L., Vasseur, A.: A parabolic problem with a fractional time derivative. Arch. Ration. Mech. Anal. 221(2), 603–630 (2016)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Allen, M., Caffarelli, L., Vasseur, A.: Porous medium flow with both a fractional potential pressure and fractional time derivative. Chin. Ann. Math. Ser. B 38(1), 45–82 (2017)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Anna, S., Fomin, S.A., Chugunov, V.A., Niibori, Y., Hashida, T.: Fractional diffusion modeling of heat transfer in porous and fractured media. Int. J. Heat Mass Transf. 103, 611–618 (2016)CrossRefGoogle Scholar
  5. 5.
    Arisawa, M.: A remark on the definitions of viscosity solutions for the integro-differential equations with Lévy operators. J. Math. Pures Appl. (9) 89(6), 567–574 (2008)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J.J.: Fractional calculus. Models and numerical methods. Second edition [of MR2894576]. Series on Complexity, Nonlinearity and Chaos, 5. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2017)Google Scholar
  7. 7.
    Bardi, M., Capuzzo-Dolcetta, I.: Optimal control and viscosity solutions of Hamilton–Jacobi–Bellman equations. With appendices by Maurizio Falcone and Pierpaolo Soravia. Systems & Control: Foundations & Applications. Birkhäuser Boston, Inc., Boston, MA (1997)Google Scholar
  8. 8.
    Barles, G.: Nonlinear Neumann boundary conditions for quasilinear degenerate elliptic equations and applications. J. Differ. Equ. 154(1), 191–224 (1999)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Barles, G., Imbert, C.: Second-order elliptic integro-differential equations: viscosity solutions’ theory revisited. Ann. Inst. H. Poincaré Anal. Non Linéaire 25(3), 567–585 (2008)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Barles, G., Ishii, H., Mitake, H.: On the large time behavior of solutions of Hamilton–Jacobi equations associated with nonlinear boundary conditions. Arch. Ration. Mech. Anal. 204(2), 515–558 (2012)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Barles, G., Mitake, H.: A PDE approach to large-time asymptotics for boundary-value problems for nonconvex Hamilton–Jacobi equations. Commun. Partial. Differ. Equ. 37(1), 136–168 (2012)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Cannarsa, P., Sinestrari, C.: Semiconcave functions, Hamilton–Jacobi equations, and optimal control. Progress in Nonlinear Differential Equations and their Applications, 58. Birkhäuser Boston, Inc., Boston, MA (2004)Google Scholar
  13. 13.
    Caputo, M.: Linear models of dissipation whose \(Q\) is almost frequency independent-II. Reprinted from Geophys. J. R. Astr. Soc. 13 (1967), no. 5, 529–539. Fract. Calc. Appl. Anal. 11 (2008), no. 1, 4–14Google Scholar
  14. 14.
    Chen, Y.G., Giga, Y., Goto, S.: Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations. J. Differ. Geom. 33(3), 749–786 (1991)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Chen, Y.G., Giga, Y., Goto, S.: Remarks on viscosity solutions for evolution equations. Proc. Jpn. Acad. Ser. A Math. Sci. 67(10), 323–328 (1991)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Crandall, M.G., Ishii, H.: The maximum principle for semicontinuous functions. Differ. Integ. Equ. 3(6), 1001–1014 (1990)MathSciNetMATHGoogle Scholar
  17. 17.
    Crandall, M.G., Ishii, H., Lions, P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. (N.S.) 27(1), 1–67 (1992)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Crandall, M.G., Kocan, M., Lions, P.L., Święch, A.: Existence results for boundary problems for uniformly elliptic and parabolic fully nonlinear equations. Electron. J. Differ. Equ. 1999(24), 1–20 (1999)MathSciNetMATHGoogle Scholar
  19. 19.
    Crandall, M.G., Lions, P.-L.: Viscosity solutions of Hamilton–Jacobi equations. Trans. Am. Math. Soc. 277(1), 1–42 (1983)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Demengel, F.: Existence’s results for parabolic problems related to fully non linear operators degenerate or singular. Potential Anal. 35(1), 1–38 (2011)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Diethelm, K.: The analysis of fractional differential equations. An application-oriented exposition using differential operators of Caputo type. Lecture Notes in Mathematics, 2004. Springer, Berlin (2010)Google Scholar
  22. 22.
    Diethelm, K., Ford, N.J., Freed, A.D., Luchko, Y.: Algorithms for the fractional calculus: a selection of numerical methods. Comput. Methods Appl. Mech. Eng. 194(6–8), 743–773 (2005)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Foufoula-Georgiou, E., Ganti, V., Dietrich, W.: A nonlocal theory of sediment transport on hillslopes. J. Geophys. Res. 115, F2 (2010)Google Scholar
  24. 24.
    Giga, Y.: Surface Evolution Equations: A Level Set Approach. Monographs in Mathematics, vol. 99. Birkhäuser Verlag, Basel (2006)MATHGoogle Scholar
  25. 25.
    Giga, Y., Namba, T.: Well-posedness of Hamilton–Jacobi equations with Caputo’s time fractional derivative. Commun. Partial Differ. Equ. 42(7), 1088–1120 (2017)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Gorenflo, R., Luchko, Y., Yamamoto, M.: Time-fractional diffusion equation in the fractional Sobolev spaces. Fract. Calc. Appl. Anal. 18(3), 799–820 (2015)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Hernández, E., O’Regan, D., Balachandran, K.: On recent developments in the theory of abstract differential equations with fractional derivatives. Nonlinear Anal. 73(10), 3462–3471 (2010)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Ishii, H.: Perron’s method for Hamilton–Jacobi equations. Duke Math. J. 55(2), 369–384 (1987)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Ishii, H.: On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic PDEs. Commun. Pure Appl. Math. 42(1), 15–45 (1989)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Ishii, H., Lions, P.-L.: Viscosity solutions of fully nonlinear second-order elliptic partial differential equations. J. Differ. Equ. 83(1), 26–78 (1990)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Jacob, N., Knopova, V.: Fractional derivatives and fractional powers as tools in understanding Wentzell boundary value problems for pseudo-differential operators generating Markov processes. Fract. Calc. Appl. Anal. 8(2), 91–112 (2005)MathSciNetMATHGoogle Scholar
  32. 32.
    Jakobsen, E.R., Karlsen, K.H.: A ”maximum principle for semicontinuous functions” applicable to integro-partial differential equations. NoDEA Nonlinear Differ. Equ. Appl. 13(2), 137–165 (2006)MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies. Elsevier, Amsterdam (2006)Google Scholar
  34. 34.
    Klages, R., Radons, G., Sokolov, I.M.: Anomalous Transport: Foundations and Applications, WILEY-VCH Verlag GmbH & Co., WeinheimGoogle Scholar
  35. 35.
    Koike, S.: A beginner’s guide to the theory of viscosity solutions. MSJ Memoirs, 13. Mathematical Society of Japan, Tokyo (2004)Google Scholar
  36. 36.
    Kolokoltsov, V.N., Veretennikova, M.A.: Well-posedness and regularity of the Cauchy problem for nonlinear fractional in time and space equations. Fract. Differ. Calc. 4(1), 1–30 (2014)MathSciNetGoogle Scholar
  37. 37.
    Lasry, J.-M., Lions, P.-L.: A remark on regularization in Hilbert spaces. Israel J. Math. 55(3), 257–266 (1986)MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Li, Z., Liu, Y., Yamamoto, M.: Initial-boundary value problems for multi-term time-fractional diffusion equations with positive constant coefficients. Appl. Math. Comput. 257, 381–397 (2015)MathSciNetMATHGoogle Scholar
  39. 39.
    Li, Z., Luchko, Y., Yamamoto, M.: Asymptotic estimates of solutions to initial-boundary-value problems for distributed order time-fractional diffusion equations. Fract. Calc. Appl. Anal. 17(4), 1114–1136 (2014)MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    Li, Z., Luchko, Y., Yamamoto, M.: Analyticity of solutions to a distributed order time-fractional diffusion equation and its application to an inverse problem. Comput. Math. Appl. 73(6), 1041–1052 (2017)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Li, C., Zeng, F.: Numerical Methods for Fractional Calculus. CRC Press, Boca Raton (2015)MATHGoogle Scholar
  42. 42.
    Luchko, Y.: Maximum principle for the generalized time-fractional diffusion equation. J. Math. Anal. Appl. 351(1), 218–223 (2009)MathSciNetMATHCrossRefGoogle Scholar
  43. 43.
    Luchko, Y.: Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation. Comput. Math. Appl. 59(5), 1766–1772 (2010)MathSciNetMATHCrossRefGoogle Scholar
  44. 44.
    Luchko, Y.: Initial-boundary problems for the generalized multi-term time-fractional diffusion equation. J. Math. Anal. Appl. 374(2), 538–548 (2011)MathSciNetMATHCrossRefGoogle Scholar
  45. 45.
    Mou, C.: Perron’s method for nonlocal fully nonlinear equations. Anal. PDE 10(5), 1227–1254 (2017)MathSciNetMATHCrossRefGoogle Scholar
  46. 46.
    Mou, C., Święch, A.: Uniqueness of viscosity solutions for a class of integro-differential equations. NoDEA Nonlinear Differ. Equ. Appl. 22(6), 1851–1882 (2015)MathSciNetMATHCrossRefGoogle Scholar
  47. 47.
    Podlubny, I.: Fractional differential equations. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA (1999)Google Scholar
  48. 48.
    Prüss, J.: Evolutionary integral equations and applications. [2012] reprint of the 1993 edition. Modern Birkhäuser Classics. Birkhäuser/Springer Basel AG, Basel (1993)Google Scholar
  49. 49.
    Sakamoto, K., Yamamoto, M.: Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems. J. Math. Anal. Appl. 382(1), 426–447 (2011)MathSciNetMATHCrossRefGoogle Scholar
  50. 50.
    Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional integrals and derivatives. Theory and applications. Edited and with a foreword by S. M. Nikol’skiĭ. Translated from the 1987 Russian original. Revised by the authors. Gordon and Breach Science Publishers, Yverdon (1993)Google Scholar
  51. 51.
    Schumer, R., Meerschaert, M.M., Baeumer, B.: Fractional advection–dispersion equations for modeling transport at the Earth surface. J. Geophys. Res. 114, F4 (2009)Google Scholar
  52. 52.
    Voller, V.R.: Fractional Stefan problems. Int. J. Heat Mass Transf. 74, 269–277 (2014)CrossRefGoogle Scholar
  53. 53.
    Zacher, R.: Weak solutions of abstract evolutionary integro-differential equations in Hilbert spaces. Funkcial. Ekvac. 52(1), 1–18 (2009)MathSciNetMATHCrossRefGoogle Scholar
  54. 54.
    Zhou, Y.: Basic Theory of Fractional Differential Equations. World Scientific Publishing Co. Pte. Ltd., Hackensack (2014)MATHCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Graduate School of Mathematical SciencesThe University of TokyoTokyoJapan

Personalised recommendations