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Local Solvability of a Linear System with a Fractional Derivative in Time in a Boundary Condition

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Abstract

In this paper we analyze a linear system for the Poisson equation with a boundary condition comprising the fractional derivative in time and the right-hand sides depended on time. First, we prove existence and uniqueness of the classical solution to this problem, and provide the coercive estimates of the solution. Second, based on the obtained results we establish one-to-one solvability to a linear system of a general form in the H¨older spaces.

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References

  1. B.V. Bazaliy, Stefan problem for the Laplace equation with regard for the curvature of the free boundary. Ukr. Math. J. 40 (1997), 1465–1484.

    Article  Google Scholar 

  2. B.V. Bazaliy, A. Friedman, The Hele-Shaw problem with surface tension in a half-plane: a model problem. J. Diff. Equations 216 (2005), 387–438.

    Article  MathSciNet  Google Scholar 

  3. B.V. Bazaliy, N. Vasil’eva, On the solvability of the Hele-Shaw model problem in weighted Hölder spaces in a plane angle. Ukrainian Math. J. 52 (2000), 1647–1660.

    Article  MathSciNet  Google Scholar 

  4. B.V. Bazaliy, N. Vasylyeva, The two-phase Hele-Shaw problem with a nonregular initial interface and without surface tension. J. Math. Phys. Anal. Geom. 10, No 1 (2014), 3–43.

    MathSciNet  MATH  Google Scholar 

  5. J.-P. Bouchard, A. Georges, Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications. Phys. Rep. 195 (1990), 127–293.

    Article  MathSciNet  Google Scholar 

  6. G. Drazer, D.H. Zanette, Experimental evidence of power-law trapping-time distributions in porous media. Phys. Rev. E 60 (1999), 5858–5864.

    Article  Google Scholar 

  7. A. Erdélyi et al. (Eds.), Higher Transcendental Functions, Vol. 3. Mc Graw-Hill, New York (1955).

    MATH  Google Scholar 

  8. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Math. Studies, 204, Elsevier Science B.V., Amsterdam (2006).

    MATH  Google Scholar 

  9. M. Kirane, N. Tatar, Nonexistence of local and global solutions of an elliptic systems with time-fractional dynamical boundary conditions. Siberian Math. J. 48 (2007), 477–488.

    Article  MathSciNet  Google Scholar 

  10. J. Klafter, G. Zumofen, M.F. Shlesinger, In: F. Mallamace, H.E. Stanley (Eds.), The Physics of Complex Systems. IOS Press, Amsterdam (1997).

  11. A.N. Kochubei, Fractional-parabolic systems. Potential Analysis 37, No 1 (2012), 1–30.

    Article  MathSciNet  Google Scholar 

  12. M. Krasnoschok, N. Vasylyeva, Existence and uniqueness of the solutions for some initial-boundary value problems with the fractional dynamic boundary condition. Inter. J. PDE 2013 (2013), Article ID 796430, 20p.

  13. M. Krasnoschok, N. Vasylyeva, On a nonclassical fractional boundary-value problem for the Laplace operator. J. Diff. Equations 257, No 6 (2014), 1814–1839.

    Article  MathSciNet  Google Scholar 

  14. M. Krasnoschok, N. Vasylyeva, On local solvability of the two-dimensional Hele-Shaw problem with a fractional derivative in time. Math. Trudy 17, No 2 (2014), 102–131.

    Google Scholar 

  15. M. Krasnoschok, N. Vasylyeva, On a solvability of a nonlinear fractional reaction-diffusion system in the Hölder spaces. J. Nonlinear Studies 20, No 4 (2013), 591–621.

    MATH  Google Scholar 

  16. O.A. Ladyzhenskaia, V.A. Solonnikov, N.N. Ural’tseva, Linear and Quasilinear Parabolic Equations. Academic Press, New York (1968).

    Book  Google Scholar 

  17. O.A. Ladyzhenskaia, N.N. Ural’tseva, Linear and Quasilinear Elliptic Equations. Academic Press, New York (1968).

    Google Scholar 

  18. B.-T. Liu, J.-P. Hsu, Theoretical analysis on diffusional release from ellipsoidal drug delivery devices. Chem. Eng. Sci. 61 (2006), 1748–1752.

    Article  Google Scholar 

  19. J.Y. Liu, M. Xu, S. Wang, Analytical solutions to the moving boundary problems with space-time-fractional derivatives in drug release devices. J. Phys. A: Math, and Theor. 40 (2007), 12131–12141.

    Article  MathSciNet  Google Scholar 

  20. A. Lunardi, Analytic semigroups and optimal regularity in parabolic problems. In: Progress in Nonlinear Differential Equations and Their Applications 16, Birkhäuser Verlag, Basel (1995).

  21. F. Mainardi, Fractional calculus: some basic problems in continuum and statistical mechanics. In: A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, Springer-Verlag, New York (1997), 291–348.

    Chapter  Google Scholar 

  22. R. Metzler, J. Klafter, Boundary value problems for fractional diffusion equations. Physica A 278 (2000), 107–125.

    Article  MathSciNet  Google Scholar 

  23. G.M. Mophou, G.M. N’Guérékata, On a class of fractional differential equations in a Sobolev space. Applicable Analysis 91 (2012), 15–34.

    Article  MathSciNet  Google Scholar 

  24. P.B. Mucha, On the Stefan problem with surface tension in the Lp framework. Adv. Diff. Equations 10, No 8 (2005), 861–900.

    MATH  Google Scholar 

  25. J.A. Ochoa-Tapia, F.J. Valdes-Parada, J. Alvarez-Ramirez, A fractional-order Darcy’s law. Physica A 374 (2007), 1–14.

    Article  Google Scholar 

  26. A.V. Pskhu, Partial Differential Equations of the Fractional Order. Nauka, Moscow, 2005 (in Russian).

    MATH  Google Scholar 

  27. A.V. Pskhu, The fundamental solution of a diffusion-wave equation of fractional order. Izvestia RAN 73 (2009), 141–182 (in Russian).

    MathSciNet  MATH  Google Scholar 

  28. K. Ritchie, X.-Y. Shan, J. Kondo, K. Iwasawa, T. Fujiwara, A. Kusumi, Detection of non-Brownian diffusion in the cell membrance in single molecule tracking. Biophys. J. 88 (2005), 2266–2277.

    Article  Google Scholar 

  29. S. Roscani, E. Santillan Marcus, A new equivalence of Stefan’s problems for the time-fractional diffusion equation. Fract. Calc. Appl. Anal. 17, No 2 (2014), 371–381; DOI: 10.2478/sl3540-014-0175-3; http://www.degruyter.eom/view/j/fca.2014.17.issue-2/sl3540-014-0175-3/sl3540-014-0175-3.xmlview/j/fca.2014.17.issue-2/sl3540-014-0175-3/sl3540-014-0175-3.xml.

    Article  MathSciNet  Google Scholar 

  30. S. Roscani, A generalization of the Hopfs lemma for the 1-D moving-boundary problem for the fractional diffusion equation and its application to a fractional free boundary problem. arXiv: 1502.01209.

  31. K. Sakamoto, M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems. J. Math. Anal. Appl. 382 (2011), 426–447.

    Article  MathSciNet  Google Scholar 

  32. V.A. Solonnikov, Estimates for the solution of the second initial-boundary value problem for the Stokes system in spaces of functions with Hölder-continuous derivatives with respect to the space variables. J. Math. Sci. 109, No 5 (2002), 1997–2017.

    Article  MathSciNet  Google Scholar 

  33. N. Vasylyeva, On a local solvability of the multidimensional Muskat problem with a fractional derivative in time on the boundary condition. Fract. Differ. Calc. 4, No 2 (2014), 89–124.

    Article  MathSciNet  Google Scholar 

  34. N. Vasylyeva, L. Vynnytska, On a multidimensional moving boundary problem governed by anomalous diffusion: analytical and numerical study. Nonlinear Differ. Equ. Appl. NoDEA, Dec. 2014; DOI: 10.1007/s00030-014-0295-9.

  35. V.R. Voller, An exact solution of a limit case Stefan problem governed by a fractional diffusion equation. Internat. J. Heat and Mass Transf. 53 (2010), 5622–5625.

    Article  Google Scholar 

  36. V.R. Voller, F. Falcini, R. Garra, Fractional Stefan problems exibiting lumped and distributed latent-heat memory effects. Phys. Rew. E 87 (2013), 042401.

  37. E. Weeks, D. Weitz, Subdiffusion and the cage effect studied near the colloidal glass transition. Chem. Phys. 284 (2002), 361–367.

    Article  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, National Academy of Sciences of Ukraine, Tereschenkivska, str. 3, Kiev - 4, 01601, Ukraine

    Nataliya Vasylyeva

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  1. Nataliya Vasylyeva
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Correspondence to Nataliya Vasylyeva.

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Vasylyeva, N. Local Solvability of a Linear System with a Fractional Derivative in Time in a Boundary Condition. FCAA 18, 982–1005 (2015). https://doi.org/10.1515/fca-2015-0058

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  • DOI: https://doi.org/10.1515/fca-2015-0058

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