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Fractional abstract Cauchy problem on complex interpolation scales

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Published/Copyright: September 11, 2020

Abstract

An fractional abstract Cauchy problem generated by a sectorial operator is investigated. An inequality of coercivity type for its solution with respect to a complex interpolation scale generated by a sectorial operator with the same parameters is established. An application to differential parabolic initial-boundary value problems in bounded domains with a fractional time derivative is shown.

Acknowledgements

The authors would like to thank referees for their valuable comments, which helped to correct inaccuracies and improve the manuscript.

References

[1] B. Baeumer, M.M. Meerschaert, Stochastic solutions for fractional Cauchy problems. Fract. Calc. Appl. Anal. 4, No 4 (2001), 481–500.Search in Google Scholar

[2] B. Baeumer, S. Kurita, M.M. Meerschaert, Inhomogeneous fractional diffusion equations. Fract. Calc. Appl. Anal. 8, No 4 (2005), 371–386.Search in Google Scholar

[3] E. Bazhlekova, The abstract Cauchy problem for fractional evolution equations. Fract. Calc. Appl. Anal. 1, No 3 (1998), 255–270.Search in Google Scholar

[4] Ph. Clément, H.J. Heijmans, S. Angenent, C.J. van Duljn, B. de Pagter. One-parameter Semigroups. North-Holland, Amsterdam (1987).Search in Google Scholar

[5] Ph. Clément, G. Gripenberg, S-O. Londen, Schauder estimates for equations with fractional derivatives. Trans. AMS352, No 5 (2000), 2239–2260.10.1090/S0002-9947-00-02507-1Search in Google Scholar

[6] C. Cuevas, C. Lizama, Almost automorphic solutions to a class of semilinear fraction differential equations. Appl. Math. Lett. 21 (2008), 1315–1319.10.1016/j.aml.2008.02.001Search in Google Scholar

[7] G. Da Prato, P. Grisvard, Equations d’évolution abstraites non linéaires de type parabolique. Ann. Mat. Pura. Appl. IV120 (1979), 329–396.10.1007/BF02411952Search in Google Scholar

[8] G. Da Prato, M. Iannelli, Existennce and regularity for a class of integrodifferential equations of parabolic type. J. Math. Anal. Appl. 112 (1985), 36–55.10.1016/0022-247X(85)90275-6Search in Google Scholar

[9] S.D. Eidelman, S.D. Ivasyshen, A.N. Kochubei, Analytic Methods in the Theory of Differential and Pseudo-Differential Equations of Parabolic Type. Birkhauser, Basel-Boston-Berlin (2004).10.1007/978-3-0348-7844-9Search in Google Scholar

[10] D. Guidetti, On maximal regularity for abstract parabolic problems with fractional time derivative. Mediterr. J. Math. (2019), 16–40.10.1007/s00009-019-1309-ySearch in Google Scholar

[11] A.V. Glushak, On the problem of Cauchy for the inhomogeneous abstract differential equation with fractional derivative. Herald of the Voronezh Univ. Ser. Phys. Math. No 1 (2002), 121–123.Search in Google Scholar

[12] M. Haase, Spectral mapping theorems for holomorphic functional calculi. J. London Math. Soc. 71, No 2 (2005), 723–739.10.1112/S0024610705006538Search in Google Scholar

[13] D. Henry, Geometric Theory of Semilinear Parabolic Equations. Lecture Notes Math. 840. Springer, Berlin-Heidelberg (1981).10.1007/BFb0089647Search in Google Scholar

[14] A. Lopushansky, Abstract parabolic Cauchy problem in complex interpolation scales. Differ. Equ. 46, No 12 (2010), 1799–1804.10.1134/S0012266110120141Search in Google Scholar

[15] A. Lopushansky, The Cauchy problem for an equation with fractional derivatives in Bessel potential spaces. Sib. Math. J. 55, No 6 (2014), 1089–1097.10.1134/S0037446614060111Search in Google Scholar

[16] C. Martínez Carracedo, M. Sanz Alix, The Theory of Fractional Powers of Operators. North-Holland, Amsterdam, (2001).Search in Google Scholar

[17] R. Seeley, Interpolation in Lp with boundary conditions. Studia Math. 44 (1972), 47-66.10.4064/sm-44-1-47-60Search in Google Scholar

[18] N. Tanabe, Equations of Evolution. Pitman, London (1979).Search in Google Scholar

[19] H. Triebel, Interpolation Theory. Function Spaces. Differential Operators. North-Holland, Amsterdam-New York-Oxford (1978).Search in Google Scholar

[20] V.S. Vladimirov, Methods of the Theory of Generelized Functions. Taylor & Francis, NY-London (2002).10.1201/9781482288162Search in Google Scholar

[21] G. Samorodnitsky, M. Taqqu, Stable non-Gaussian Random Processes. Chapman and Hall, New York (1994).Search in Google Scholar

[22] A. Saichev, G. Zaslavsky, Fractional kinetic equations: solutions and applications. Chaos. 7, No 4 (1997), 759–764.10.1063/1.166272Search in Google Scholar PubMed

Received: 2019-02-19
Revised: 2020-07-30
Published Online: 2020-09-11
Published in Print: 2020-08-26

© 2020 Diogenes Co., Sofia

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