Abstract
We consider initial value problems for abstract evolution equations with fractional time derivative. Concerning the Caputo derivative \(\mathbb {D}^\alpha u\), we show that certain assumptions, which are known to be sufficient to get a unique solution with a prescribed regularity, are also necessary. So we establish a maximal regularity result. We consider similar problems with the Riemann–Liouville derivative \(\partial ^\alpha u\). Here, we give a complete proof (necessity and sufficiency of the assumptions) of the corresponding maximal regularity results.
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References
Adams, R.A., Fournier, J.F.: “Sobolev spaces”, Pure and Applied Mathematics Series, vol. 140. Academi Press, Cambridge (2003)
Bazhlekova, E.: Fractional Evolution Equations in Banach Spaces. Technische Universiteit Eindhoven, Eindhoven (2001)
Bazhlekova, E.: Strict \(L^p\) solutions for fractional evolution equations. Fract. Calc. Appl. Anal. 5, 427–436 (2002)
Clement, P., Gripenberg, G., Londen, S.-O.: Schauder estimates for equations with fractional derivatives. Trans. Am. Math. Soc. 352, 2239–2260 (2000)
Clement, P., Gripenberg, G., Londen, S.-O.: Regularity Properties of Solutions of Fractional Evolution Equations. Lecture Notes in Pure and Applied Mathematics, vol. 215. Dekker, New York (2001)
Clement, P., Londen, S.-O., Simonett, G.: Quasilinear evolutionary equations and continuous interpolation spaces. J. Differ. Eq. 196, 418–447 (2004)
Da Prato, G., Grisvard, P.: Sommes d’opérateurs linéaires et équations différentielles opérationelles. J. Math. Pure Appl. 54, 305–387 (1975)
Fabrizio, M., Giorgi, C., Morro, A.: Modeling of heat conduction via fractional derivatives. Heat Mass Transf. 53(9), 2785–2797 (2017)
Guidetti, D.: On interpolation with boundary conditions. Math. Z. 207, 439–460 (1991)
Guidetti, D.: On maximal regularity for the Cauchy–Dirichlet mixed parabolic problem with fractional time derivative. ArXiv: 1707.02093v1 (2017)
Keyantuo, V., Lizama, C.: Hölder continuous solutions for integro-differential equations and maximal regularity. J. Differ. Eq. 230, 634–660 (2006)
Lunardi, A.: Analytic Semigroups and Optimal Regularity in Parabolic Problems. Progress in Nonlinear Differential Equations and their Applications. Birkhäuser, Bern (1995)
Podlubny, I.: Fractional Differential Equations, Mathematics in Science and Engineering, vol. 198. Academic Press, Cambridge (1999)
Ponce, R.: Hölder continuous solutions for fractional differential equations and maximal regularity. J. Differ. Eq. 255, 3284–3304 (2013)
Stewart, H.B.: Generation of analytic semigroups by strongly elliptic operators under general boundary conditions. Trans. Am. Math. Soc. 259, 299–310 (1980)
Tanabe, H.: Equations of Evolution. Pitman, New Jersey (1979)
Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators, vol. 18. North-Holland Pub. Co., Amsterdam, New York (1978)
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Guidetti, D. On Maximal Regularity for Abstract Parabolic Problems with Fractional Time Derivative. Mediterr. J. Math. 16, 40 (2019). https://doi.org/10.1007/s00009-019-1309-y
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DOI: https://doi.org/10.1007/s00009-019-1309-y