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.
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.Search in 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.Search in Google Scholar
[3] B.V. Bazaliy, N. Vasil’eva, On the solvability of the Hele-Shaw model problem in weighted H¨older spaces in a plane angle. Ukrainian Math. J. 52 (2000), 1647-1660.Search in 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.Search in 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.Search in Google Scholar
[6] G. Drazer, D.H. Zanette, Experimental evidence of power-law trappingtime distributions in porous media. Phys. Rev. E 60 (1999), 5858-5864.Search in Google Scholar
[7] A. Erd´elyi at al. (Eds.), Higher Transcendental Functions, Vol. 3. Mc Graw-Hill, New York (1955).Search in 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).Search in 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.Search in 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).Search in Google Scholar
[11] A.N. Kochubei, Fractional-parabolic systems. Potential Analysis 37, No 1 (2012), 1-30.Search in 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.10.1155/2013/796430Search in Google Scholar
[13] M. Krasnoschok, N. Vasylyeva, On a nonclassical fractional boundaryvalue problem for the Laplace operator. J. Diff. Equations 257, No 6 (2014), 1814-1839.Search in Google Scholar
[14] M. Krasnoschok, N. Vasylyeva, On local solvability of the twodimensional Hele-Shaw problem with a fractional derivative in time. Math. Trudy 17, No 2 (2014), 102-131.Search in Google Scholar
[15] M. Krasnoschok, N. Vasylyeva, On a solvability of a nonlinear fractional reaction-diffusion system in the H¨older spaces. J. Nonlinear Studies 20, No 4 (2013), 591-621.Search in Google Scholar
[16] O.A. Ladyzhenskaia, V.A. Solonnikov, N.N. Ural’tseva, Linear and Quasilinear Parabolic Equations. Academic Press, New York (1968).Search in Google Scholar
[17] O.A. Ladyzhenskaia, N.N. Ural’tseva, Linear and Quasilinear Elliptic Equations. Academic Press, New York (1968).Search in 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.Search in 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.Search in Google Scholar
[20] A. Lunardi, Analytic semigroups and optimal regularity in parabolic problems. In: Progress in Nonlinear Differential Equations and Their Applications 16, Birkh¨auser Verlag, Basel (1995).Search in Google Scholar
[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. 10.1007/978-3-7091-2664-6_7Search in Google Scholar
[22] R. Metzler, J. Klafter, Boundary value problems for fractional diffusion equations. Physica A 278 (2000), 107-125.Search in Google Scholar
[23] G.M. Mophou, G.M. N’Gu´er´ekata, On a class of fractional differential equations in a Sobolev space. Applicable Analysis 91 (2012), 15-34.Search in 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.Search in 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.Search in Google Scholar
[26] A.V. Pskhu, Partial Differential Equations of the Fractional Order. Nauka, Moscow, 2005 (in Russian).Search in 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).10.1070/IM2009v073n02ABEH002450Search in 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.Search in 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/s13540-014-0175-3; http://www.degruyter.com/view/j/fca.2014.17.issue-2/s13540-014-0175-3/s13540-014-0175-3.xml.10.2478/s13540-014-0175-3Search in Google Scholar
[30] S. Roscani, A generalization of the Hopf’s lemma for the 1-D movingboundary problem for the fractional diffusion equation and its application to a fractional free boundary problem. arXiv:1502.01209.Search in Google Scholar
[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.Search in Google Scholar
[32] V.A. Solonnikov, Estimates for the solution of the second initialboundary value problem for the Stokes system in spaces of functions with H¨older-continuous derivatives with respect to the space variables. J. Math. Sci. 109, No 5 (2002), 1997-2017.Search in 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.Search in 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/s00030-014-0295-9.10.1007/s00030-014-0295-9Search in Google Scholar
[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. Search in Google Scholar
© Diogenes Co., Sofia - frontmatter and editorial