메뉴 건너뛰기




Volumn 257, Issue 6, 2014, Pages 1814-1839

On a nonclassical fractional boundary-value problem for the Laplace operator

Author keywords

Caputo derivative; Coercive estimates; Elliptic equations; Fractional dynamic boundary condition

Indexed keywords


EID: 84903765327     PISSN: 00220396     EISSN: 10902732     Source Type: Journal    
DOI: 10.1016/j.jde.2014.05.022     Document Type: Article
Times cited : (26)

References (49)
  • 1
    • 0039229292 scopus 로고    scopus 로고
    • A Fujita-type theorem for the Laplace equation with a dynamical boundary condition
    • Amann H., Fila M. A Fujita-type theorem for the Laplace equation with a dynamical boundary condition. Acta Math. Univ. Comenian. 1997, LXVI:321-328.
    • (1997) Acta Math. Univ. Comenian. , vol.66 , pp. 321-328
    • Amann, H.1    Fila, M.2
  • 2
    • 33845481306 scopus 로고    scopus 로고
    • A degenerate elliptic-parabolic problem with nonlinear dynamical boundary conditions
    • Andreu F., Igbida N., Mazon J.M., Toledo J. A degenerate elliptic-parabolic problem with nonlinear dynamical boundary conditions. Interfaces Free Bound. 2006, 4:447-479.
    • (2006) Interfaces Free Bound. , vol.4 , pp. 447-479
    • Andreu, F.1    Igbida, N.2    Mazon, J.M.3    Toledo, J.4
  • 4
    • 84903795160 scopus 로고    scopus 로고
    • On a proof of the classical solvability of the Hele-Shaw problem with a free boundary
    • Bazaliy B.V. On a proof of the classical solvability of the Hele-Shaw problem with a free boundary. Ukrainian Math. J. 1998, 50:1659-1670.
    • (1998) Ukrainian Math. J. , vol.50 , pp. 1659-1670
    • Bazaliy, B.V.1
  • 5
    • 24144447338 scopus 로고    scopus 로고
    • The Hele-Shaw problem with surface tension in a half-plane
    • Bazaliy B.V., Friedman A. The Hele-Shaw problem with surface tension in a half-plane. J. Differential Equations 2005, 216:439-469.
    • (2005) J. Differential Equations , vol.216 , pp. 439-469
    • Bazaliy, B.V.1    Friedman, A.2
  • 6
    • 24144500573 scopus 로고    scopus 로고
    • On the solvability of the Hele-Shaw model problem in weighted Hölder spaces in a plane angle
    • Bazaliy B.V., Vasil'eva N. On the solvability of the Hele-Shaw model problem in weighted Hölder spaces in a plane angle. Ukrainian Math. J. 2000, 52:1647-1660.
    • (2000) Ukrainian Math. J. , vol.52 , pp. 1647-1660
    • Bazaliy, B.V.1    Vasil'eva, N.2
  • 7
    • 0011140206 scopus 로고    scopus 로고
    • An abstract approximate controllability result and applications to elliptic and parabolic system with dynamic boundary conditions
    • Bejenaru I., Diaz J.I., Vrabie I.I. An abstract approximate controllability result and applications to elliptic and parabolic system with dynamic boundary conditions. Electron. J. Differential Equations 2001, 50:1-19.
    • (2001) Electron. J. Differential Equations , vol.50 , pp. 1-19
    • Bejenaru, I.1    Diaz, J.I.2    Vrabie, I.I.3
  • 8
    • 0040307478 scopus 로고
    • Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications
    • Bouchard J.-P., Georges A. Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications. Phys. Rep. 1990, 195:127-293.
    • (1990) Phys. Rep. , vol.195 , pp. 127-293
    • Bouchard, J.-P.1    Georges, A.2
  • 9
    • 0000377342 scopus 로고    scopus 로고
    • Experimental evidence of power-law trapping-time distributions in porous media
    • Drazer G., Zanette D.H. Experimental evidence of power-law trapping-time distributions in porous media. Phys. Rev. E 1999, 60:5858-5864.
    • (1999) Phys. Rev. E , vol.60 , pp. 5858-5864
    • Drazer, G.1    Zanette, D.H.2
  • 11
    • 84860915104 scopus 로고
    • Smooth solutions of nonlinear elliptic systems with dynamic boundary conditions
    • Escher J. Smooth solutions of nonlinear elliptic systems with dynamic boundary conditions. Lect. Notes Pure Appl. Math. 1994, vol. 155:173-183.
    • (1994) Lect. Notes Pure Appl. Math. , vol.155 , pp. 173-183
    • Escher, J.1
  • 12
    • 0031478022 scopus 로고    scopus 로고
    • Classical solutions of multidimensional Hele-Shaw models
    • Escher J., Simonett G. Classical solutions of multidimensional Hele-Shaw models. SIAM J. Math. Anal. 1997, 28:1028-1047.
    • (1997) SIAM J. Math. Anal. , vol.28 , pp. 1028-1047
    • Escher, J.1    Simonett, G.2
  • 13
    • 0031257438 scopus 로고    scopus 로고
    • Global solutions of the Laplace equation with a nonlinear dynamical boundary condition
    • Fila M., Quittner P. Global solutions of the Laplace equation with a nonlinear dynamical boundary condition. Math. Methods Appl. Sci. 1997, 20:1325-1333.
    • (1997) Math. Methods Appl. Sci. , vol.20 , pp. 1325-1333
    • Fila, M.1    Quittner, P.2
  • 14
    • 0001278174 scopus 로고
    • The initial value problem for the linearized equations of water-waves
    • Friedman A., Shinbort M. The initial value problem for the linearized equations of water-waves. J. Math. Mech. 1967, 17:107-180.
    • (1967) J. Math. Mech. , vol.17 , pp. 107-180
    • Friedman, A.1    Shinbort, M.2
  • 15
    • 34547830478 scopus 로고    scopus 로고
    • Estimates for solution to model problems connected with quasistationary approximation for the Stefan problem
    • Frolova E.V. Estimates for solution to model problems connected with quasistationary approximation for the Stefan problem. J. Math. Sci. 2004, 124:5070-5083.
    • (2004) J. Math. Sci. , vol.124 , pp. 5070-5083
    • Frolova, E.V.1
  • 16
    • 2142789368 scopus 로고
    • On the linear theory of gravity waves: the theorem of existence and uniqueness
    • Garipov R.M. On the linear theory of gravity waves: the theorem of existence and uniqueness. Arch. Ration. Mech. Anal. 1967, 14:352-362.
    • (1967) Arch. Ration. Mech. Anal. , vol.14 , pp. 352-362
    • Garipov, R.M.1
  • 17
    • 44649089038 scopus 로고    scopus 로고
    • Boundary value problems for fractional diffusion-wave equation
    • Gejji V.D., Jafari H. Boundary value problems for fractional diffusion-wave equation. Aust. J. Math. Anal. Appl. 2006, 3:1-8.
    • (2006) Aust. J. Math. Anal. Appl. , vol.3 , pp. 1-8
    • Gejji, V.D.1    Jafari, H.2
  • 19
    • 84903771370 scopus 로고    scopus 로고
    • Howison S.D. http://www.maths.ox.ac.uk/howison/Hele-Shaw.
    • Howison, S.D.1
  • 20
    • 0036982746 scopus 로고    scopus 로고
    • A degenerate diffusion problem with dynamical boundary conditions
    • Igibida N., Kirane M. A degenerate diffusion problem with dynamical boundary conditions. Math. Ann. 2002, 323:377-396.
    • (2002) Math. Ann. , vol.323 , pp. 377-396
    • Igibida, N.1    Kirane, M.2
  • 21
    • 34547120408 scopus 로고    scopus 로고
    • Fractional calculus of the generalized Wright functions
    • Kilbas A. Fractional calculus of the generalized Wright functions. Fract. Calc. Appl. Anal. 2005, 8:113-126.
    • (2005) Fract. Calc. Appl. Anal. , vol.8 , pp. 113-126
    • Kilbas, A.1
  • 22
    • 77956684069 scopus 로고    scopus 로고
    • Theory and Applications of Fractional Differential Equations
    • Elsevier Science B.V., Amsterdam
    • Kilbas A.A., Srivastava H.M., Trujillo J.J. Theory and Applications of Fractional Differential Equations. North-Holland Math. Stud. 2006, vol. 204. Elsevier Science B.V., Amsterdam.
    • (2006) North-Holland Math. Stud. , vol.204
    • Kilbas, A.A.1    Srivastava, H.M.2    Trujillo, J.J.3
  • 23
    • 34347246652 scopus 로고    scopus 로고
    • Nonexistence of global solutions to an elliptic equation with dynamical boundary condition
    • Kirane M., Nabana E., Pohozaev S.I. Nonexistence of global solutions to an elliptic equation with dynamical boundary condition. Bol. Soc. Parana. Mat. 2004, 22:9-16.
    • (2004) Bol. Soc. Parana. Mat. , vol.22 , pp. 9-16
    • Kirane, M.1    Nabana, E.2    Pohozaev, S.I.3
  • 24
    • 34347207584 scopus 로고    scopus 로고
    • Nonexistence of local and global solutions of an elliptic systems with time-fractional dynamical boundary conditions
    • Kirane M., Tatar N. Nonexistence of local and global solutions of an elliptic systems with time-fractional dynamical boundary conditions. Sib. Math. J. 2007, 48:477-488.
    • (2007) Sib. Math. J. , vol.48 , pp. 477-488
    • Kirane, M.1    Tatar, N.2
  • 26
    • 84862237506 scopus 로고    scopus 로고
    • Fractional-parabolic systems
    • Kochubei A.N. Fractional-parabolic systems. Potential Anal. 2012, 37(1):1-30.
    • (2012) Potential Anal. , vol.37 , Issue.1 , pp. 1-30
    • Kochubei, A.N.1
  • 27
    • 84903768974 scopus 로고    scopus 로고
    • Existence and uniqueness of the solutions for some initial-boundary value problems with the fractional dynamic boundary condition
    • Article ID 796430
    • Krasnoschok M., Vasylyeva N. Existence and uniqueness of the solutions for some initial-boundary value problems with the fractional dynamic boundary condition. Inter. J. Partial Differential Equations 2013, 2013. Article ID 796430, 20 pp.
    • (2013) Inter. J. Partial Differential Equations , vol.2013 , pp. 20
    • Krasnoschok, M.1    Vasylyeva, N.2
  • 30
    • 34748841026 scopus 로고    scopus 로고
    • Analytical solutions to the moving boundary problems with space-time-fractional derivatives in drug release devices
    • Liu J.Y., Xu M., Wang S. Analytical solutions to the moving boundary problems with space-time-fractional derivatives in drug release devices. J. Phys. A 2007, 40:12131-12141.
    • (2007) J. Phys. A , vol.40 , pp. 12131-12141
    • Liu, J.Y.1    Xu, M.2    Wang, S.3
  • 31
    • 0003236729 scopus 로고
    • Analytic Semigroups and Optimal Regularity in Parabolic Problems
    • Birkhäuser Verlag, Basel
    • Lunardi A. Analytic Semigroups and Optimal Regularity in Parabolic Problems. Progr. Nonlinear Differential Equations Appl. 1995, vol. 16. Birkhäuser Verlag, Basel.
    • (1995) Progr. Nonlinear Differential Equations Appl. , vol.16
    • Lunardi, A.1
  • 32
    • 0001983732 scopus 로고    scopus 로고
    • Fractional calculus: some basic problems in continuum and statistical mechanics
    • Springer-Verlag, New York, A. Garpinteri, F. Mainardi (Eds.)
    • Mainardi F. Fractional calculus: some basic problems in continuum and statistical mechanics. Fractals and Fractional Calculus in Continuum Mechanics 1997, 291-348. Springer-Verlag, New York. A. Garpinteri, F. Mainardi (Eds.).
    • (1997) Fractals and Fractional Calculus in Continuum Mechanics , pp. 291-348
    • Mainardi, F.1
  • 33
    • 0033884660 scopus 로고    scopus 로고
    • Boundary value problems for fractional diffusion equations
    • Metzler R., Klafter J. Boundary value problems for fractional diffusion equations. Phys. A 2000, 278:107-125.
    • (2000) Phys. A , vol.278 , pp. 107-125
    • Metzler, R.1    Klafter, J.2
  • 34
    • 79960974149 scopus 로고    scopus 로고
    • On a class of fractional differential equations in a Sobolev space
    • Mophou G.M., N'Guérékata G.M. On a class of fractional differential equations in a Sobolev space. Appl. Anal. 2012, 91:15-34.
    • (2012) Appl. Anal. , vol.91 , pp. 15-34
    • Mophou, G.M.1    N'Guérékata, G.M.2
  • 36
    • 84891754187 scopus 로고
    • The Laplace transformation for linear differential equation of the fractional order
    • I. Podlubny, The Laplace transformation for linear differential equation of the fractional order, Preprint UEF-02-94, 1994.
    • (1994) Preprint UEF-02-94
    • Podlubny, I.1
  • 39
    • 65349154473 scopus 로고    scopus 로고
    • The fundamental solution of a diffusion-wave equation of fractional order
    • (in Russian)
    • Pskhu A.V. The fundamental solution of a diffusion-wave equation of fractional order. Izv. Ross. Akad. Nauk 2009, 73:141-182. (in Russian).
    • (2009) Izv. Ross. Akad. Nauk , vol.73 , pp. 141-182
    • Pskhu, A.V.1
  • 40
    • 21244494104 scopus 로고    scopus 로고
    • Detection of non-Brownian diffusion in the cell membrane in single molecule tracking
    • Ritchie K., Shan X.-Y., Kondo J., Iwasawa K., Fujiwara T., Kusumi A. Detection of non-Brownian diffusion in the cell membrane in single molecule tracking. Biophys. J. 2005, 88:2266-2277.
    • (2005) Biophys. J. , vol.88 , pp. 2266-2277
    • Ritchie, K.1    Shan, X.-Y.2    Kondo, J.3    Iwasawa, K.4    Fujiwara, T.5    Kusumi, A.6
  • 41
    • 79956068443 scopus 로고    scopus 로고
    • Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems
    • Sakamoto K., Yamamoto M. Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems. J. Math. Anal. Appl. 2011, 382:426-447.
    • (2011) J. Math. Anal. Appl. , vol.382 , pp. 426-447
    • Sakamoto, K.1    Yamamoto, M.2
  • 43
    • 52649108286 scopus 로고    scopus 로고
    • 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
    • Solonnikov V.A. 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. 2002, 109(5):1997-2017.
    • (2002) J. Math. Sci. , vol.109 , Issue.5 , pp. 1997-2017
    • Solonnikov, V.A.1
  • 44
    • 33746894529 scopus 로고    scopus 로고
    • On the Laplace equation with nonlinear dynamical boundary conditions
    • Vitillaro E. On the Laplace equation with nonlinear dynamical boundary conditions. Proc. London Math. Soc. 2006, 93:418-446.
    • (2006) Proc. London Math. Soc. , vol.93 , pp. 418-446
    • Vitillaro, E.1
  • 45
    • 77956056756 scopus 로고    scopus 로고
    • An exact solution of a limit case Stefan problem governed by a fractional diffusion equation
    • Voller V.R. An exact solution of a limit case Stefan problem governed by a fractional diffusion equation. Int. J. Heat Mass Transfer. 2010, 53:5622-5625.
    • (2010) Int. J. Heat Mass Transfer. , vol.53 , pp. 5622-5625
    • Voller, V.R.1
  • 46
    • 84876722376 scopus 로고    scopus 로고
    • Fractional Stefan problems exhibiting lumped and distributed latent-heat memory effects
    • 042401
    • Voller V.R., Falcini F., Garra R. Fractional Stefan problems exhibiting lumped and distributed latent-heat memory effects. Phys. Rew. E 2013, 87. 042401.
    • (2013) Phys. Rew. E , vol.87
    • Voller, V.R.1    Falcini, F.2    Garra, R.3
  • 47
    • 0036830173 scopus 로고    scopus 로고
    • Subdiffusion and the cage effect studied near the colloidal glass transition
    • Weeks E., Weitz D. Subdiffusion and the cage effect studied near the colloidal glass transition. Chem. Phys. 2002, 284:361-367.
    • (2002) Chem. Phys. , vol.284 , pp. 361-367
    • Weeks, E.1    Weitz, D.2
  • 48
    • 0012719785 scopus 로고
    • The generalized Bessel function of order greater than one
    • Wright E.M. The generalized Bessel function of order greater than one. Quart. J. Math. 1940, 11:36-48.
    • (1940) Quart. J. Math. , vol.11 , pp. 36-48
    • Wright, E.M.1
  • 49
    • 9544255692 scopus 로고    scopus 로고
    • Global classical solution of quasi-stationary Stefan free boundary problem
    • Yi F. Global classical solution of quasi-stationary Stefan free boundary problem. Appl. Math. Comput. 2005, 160:797-817.
    • (2005) Appl. Math. Comput. , vol.160 , pp. 797-817
    • Yi, F.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.