×

The Cauchy problem for an equation with fractional derivatives in Bessel potential spaces. (English. Russian original) Zbl 1323.35199

Sib. Math. J. 55, No. 6, 1089-1097 (2014); translation from Sib. Mat. Zh. 55, No. 6, 1334-1344 (2014).
Summary: For an equation with fractional derivatives we establish the existence of a solution to the Cauchy problem which is classical in time and belongs to Bessel potential classes in space variables.

MSC:

35R11 Fractional partial differential equations
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Eidelman S. D., Ivasyshen S. D., and Kochubei A. N., Analytical Methods in the Theory of Differential and Pseudo-Differential Equations of Parabolic Type, Birkhäuser-Verlag, Basel, Boston, and Berlin (2004). · Zbl 1062.35003
[2] Dzhrbashyan M. M., Integral Transformations and Representations of Functions in a Complex Domain [in Russian], Nauka, Moscow (1999).
[3] Anh V. V. and Leonenko N. N., “Spectral analysis of fractional kinetic equations with random data,” J. Stat. Phys., 104, No. 5/6, 1349-1387 (2001). · Zbl 1034.82044
[4] Sheng D. J., “Time- and space-fractional partial differential equations,” J. Math. Phys., 46, 13504-13511 (2005). · Zbl 1076.26006
[5] Gorenfio R., Iskenderov A., and Luchko Yu., “Mapping between solutions of fractional diffusion-wave equations,” Fract. Calc. Appl. Anal., 3, 75-86 (2000). · Zbl 1033.35161
[6] Hanyga A., “Multi-dimensional solutions for space-time-fractional diffusion equations,” Proc. R. Soc. Lond., A 458, 429-450 (2002). · Zbl 0999.60035
[7] Luchko Yu., “Fractional wave equation and damped waves,” J. Math. Phys., 54, 315051-3150516 (2013).
[8] Luchko Yu., “Multi-dimensional fractional wave equation and some properties of its fundamental solution,” E-print Arxiv: 1311.5920[math-ph]. · Zbl 1329.35335
[9] Luchko Yu. and Punzi A., “Modeling anomalous heat transport in geothermal reservoirs via fractional diffusion equations,” Int. J. Geomath., 1, 257-276 (2011). · Zbl 1301.34104
[10] Magin R. L., “Fractional calculus in bioengineering: P. 1-3,” Crit. Rev. Biomed. Engineering, 32, 1-104, 105-193, 195-377 (2004).
[11] Mainardi F., Fractional calculus and waves in linear viscoelasticity, Imperial College Press, London (2010). · Zbl 1210.26004
[12] Mainardi F., Luchko Yu., and Pagnini G., “The fundamental solution of the space-time-fractional diffusion equation,” Fract. Calc. Appl. Anal., 4, 153-192 (2001). · Zbl 1054.35156
[13] Metzler R. and Nonnenmacher T. F., “Space- and time-fractional diffusion and wave equations, fractional Fokker-Planck equations, and physical motivation,” Chem. Phys., 284, 67-90 (2002).
[14] Povstenko Yu., “Theories of thermal stresses based on space-time-fractional telegraph equations,” Computer Math. Appl., 64, 3321-3328 (2012). · Zbl 1268.74018
[15] Lopushanska H. P. and Lopushans’kyi A. O., “Space-time fractional Cauchy problem in spaces of generalized functions,” Ukrainian Math. J., 64, No. 8, 1215-1230 (2013). · Zbl 1302.35407
[16] Lopushanska H. P., and Lopushanskyj A. O., and Pasichnik E. V., “The Cauchy problem in a space of generalized functions for the equations possessing the fractional time derivative,” Siberian Math. J., 52, No. 6, 1022-1299 (2011). · Zbl 1235.35281
[17] Herrmann R., Fractional Calculus: An Introduction for Physicists, World Sci., Singapore (2011). · Zbl 1232.26006
[18] Hilfer R. (Ed.), Applications of Fractional Calculus in Physics, World Sci., Singapore (2000). · Zbl 0998.26002
[19] Klages R., Radons G., and Sokolov I. M. (Eds.), Anomalous Transport: Foundations and Applications, Wiley-VCH, Weinheim (2008).
[20] Vladimirov V. S., Generalized Functions in Mathematical Physics [in Russian], Nauka, Moscow (1979). · Zbl 0515.46034
[21] Shilov G. E., Mathematical Analysis. Second Special Course [in Russian], Nauka, Moscow (1965). · Zbl 0137.26203
[22] Kreĭn S. G. (ed.), Functional Analysis [in Russian], Nauka, Moscow (1972).
[23] Roĭtberg Ya. A., Elliptic Boundary Value Problems in Generalized Functions. I [in Russian], Chernigov Ped. Inst., Chernigov (1990).
[24] Taylor M. E., Pseudodifferential Operators [Russian translation], Mir, Moscow (1985).
[25] Kilbas A. A. and Sajgo M., H-Transforms, Chapman and Hall/CRC, Boca Raton, FL (2004). · Zbl 1056.44001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.