Solvability in Hölder space of an initial boundary value problem for the time-fractional diffusion equation
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 12 (2016) no. 1, pp. 48-77 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In this paper we consider an initial boundary value problem for the time-fractional diffusion equation with mixed boundary conditions. We prove a theorem of existence and uniqueness of solution to this problem in Hölder spaces.
@article{JMAG_2016_12_1_a2,
     author = {M. V. Krasnoschok},
     title = {Solvability in {H\"older} space of an initial boundary value problem for the time-fractional diffusion equation},
     journal = {\v{Z}urnal matemati\v{c}eskoj fiziki, analiza, geometrii},
     pages = {48--77},
     year = {2016},
     volume = {12},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JMAG_2016_12_1_a2/}
}
TY  - JOUR
AU  - M. V. Krasnoschok
TI  - Solvability in Hölder space of an initial boundary value problem for the time-fractional diffusion equation
JO  - Žurnal matematičeskoj fiziki, analiza, geometrii
PY  - 2016
SP  - 48
EP  - 77
VL  - 12
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/JMAG_2016_12_1_a2/
LA  - en
ID  - JMAG_2016_12_1_a2
ER  - 
%0 Journal Article
%A M. V. Krasnoschok
%T Solvability in Hölder space of an initial boundary value problem for the time-fractional diffusion equation
%J Žurnal matematičeskoj fiziki, analiza, geometrii
%D 2016
%P 48-77
%V 12
%N 1
%U http://geodesic.mathdoc.fr/item/JMAG_2016_12_1_a2/
%G en
%F JMAG_2016_12_1_a2
M. V. Krasnoschok. Solvability in Hölder space of an initial boundary value problem for the time-fractional diffusion equation. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 12 (2016) no. 1, pp. 48-77. http://geodesic.mathdoc.fr/item/JMAG_2016_12_1_a2/

[1] M. Caputo, “Models of Flux in Porous Media with Memory”, Water Resources Research, 36:3 (2000), 693–705 | DOI

[2] Ph. Clément, S.-O. Londen, G. Simonett, “Quasilinear Evolutionary Equations and Continuous Interpolation Spaces”, J. Diff. Eqs., 196:2 (2004), 418–447 | DOI | MR

[3] S. D. Eidelman, A. N. Kochubei, “Cauchy Problem for Fractional Diffusion Equations”, J. Diff. Eqs., 199 (2004), 211–255 | DOI | MR | Zbl

[4] H. Engler, “Strong Solutions of Quasilinear Integro-Differential Equations with Singular Kernles in Several Space Dimensions”, Electronic J. Diff. Eqs., 1995, no. 2, 1–16 | MR

[5] I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Sums, Series and Products, Fizmatgiz, M., 1963 (in Russian)

[6] J. Kemppainen, “Existence and Uniqueness of the Solution for a Time-Fractional Diffusion Equation with Robin Boundary Condition”, Abstract and Applied Analysis, 2011, Article ID 321903, 11 pp. | MR | Zbl

[7] A. A. Kilbas, “Fractional Calculus of the Generalized Wright Functions”, Fractional Calculus and Applied Analysis, 8:2 (2005), 113–126 | MR | Zbl

[8] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North Holland Mathematical studies, 204, Elsevier Science B.V., Amsterdam, 2006 | MR | Zbl

[9] A. N. Kochubei, “Fractional-Order Diffusion”, Diff. Eqs., 26 (1990), 485–492 | MR | Zbl

[10] A. N. Kochubei, “Fractional Parabolic Systems”, Potential Analysis, 37 (2012), 1–30 | DOI | MR | Zbl

[11] M. Krasnoschok, N. Vasylyieva, “On a Solvability of Nonlinear Fractional Reaction-Diffusion System in the Hölder Spaces”, Nonlinear Studies, 20:4 (2013), 591–621 | MR | Zbl

[12] O. A. Ladyzhenskaya, V. A. Solonnikov, N. N. Ural'tseva, Linear and Quasilinear Parabolic Equations, Transl. Math. Monogr., 23, AMS, Providence, RI, 1968

[13] H. P. Lopushans'ka, A. O. Lopushans'kyi, “Space-Time Fractional Cauchy Problem in Spaces of Generalized Functions”, Ukr. Math. J., 64:8 (2013), 1215–1230 | DOI | MR | Zbl

[14] R. Metzler, J. Klafter, “The Random Walk's Guide to Anomalous Diffusion: a Fractional Dynamics Approach”, Phys. Rep., 339 (2000), 1–77 | DOI | MR | Zbl

[15] G. M. Mophou, G. M. N'Guérékata, “On a Class of Fractional Differential Equations in a Sobolev Space”, Applicable Analysis, 91:1 (2012), 15–34 | DOI | MR | Zbl

[16] A. M. Nahushev, Fractional Calculus and Its Applications, Nauka, M., 2003 (in Russian)

[17] I. Podlubny, Fracrional Differential Equations, Academic Press, San-Diego, 1999 | MR

[18] R. Ponce, “Hölder Continuous Solutions for Fractional Differential Equations and Maximal Regularity”, J. Diff. Eqs., 255 (2013), 3284–3304 | DOI | MR | Zbl

[19] A. V. Pskhu, “A Fundamental Solution for a Fractional Diffusion Wave Equation”, Izvestia RAN, 73 (2009), 141–181 (in Russian) | MR

[20] A. V. Pskhu, Partial Differential Equations of the Fractional Order, Nauka, M., 2005 (in Russian) | MR

[21] 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 | DOI | MR | Zbl

[22] W. R. Schneider, W. Wyss, “Fractional Diffusion and Wave Equations”, J. Math. Phys., 30 (1989), 134–144 | DOI | MR | Zbl

[23] V. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles. Fields and Media, Higher Education Press, Beijing; Springer-Verlag, Berlin–Heidelberg, 2010 | MR | Zbl

[24] R. Zacher, Quasilinear Parabolic Problems with Nonlinear Boundary Conditions, Ph. D. Thesis, Martin-Luther-Universität, Halle–Wittenberg, 2003