On maximal regularity for the Cauchy-Dirichlet parabolic problem with fractional time derivative

We are concerned with a fractional abstract Cauchy problem for possibly degenerate equations in Banach spaces. This form of degeneration may be strong and some convenient assumptions about the involved operators are required to handle the direct problem. Moreover, we succeeded in handling related inverse problems, extending the treatment given by Alfredo Lorenzi. Some basic assumptions on the involved operators are also introduced allowing application of the real interpolation theory of Lions and Peetre. Our abstract approach improves previous results given by Favini–Yagi by using more general real interpolation spaces with indices θ , p, p ∈ ( 0 , ∞ ] instead of the indices θ , ∞. As a possible application of the abstract theorems, some examples of partial differential equations are given.

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“…In Section 6, we present our conclusions and remarks. For some related results, we refer to Guidetti [10] and Bazhlekova [11].…”

Section: Consider the Abstract Equationmentioning

confidence: 99%

Direct and Inverse Fractional Abstract Cauchy Problems

2019

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“…and Γ(β) is the Gamma function. For Riemann-Liouville derivative Dα t of orderα, we address the monograph [11] (see also [12,13]). Very recent applications concerning Caputo fractional derivative operator are also discussed in [14] by the same authors using a completely different method than Sviridyuk's group (see [15,16]).…”

Section: Introductionmentioning

confidence: 99%

Fractional Cauchy Problems for Infinite Interval Case-II

et al. 2019

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We consider fractional abstract Cauchy problems on infinite intervals. A fractional abstract Cauchy problem for possibly degenerate equations in Banach spaces is considered. This form of degeneration may be strong and some convenient assumptions about the involved operators are required to handle the direct problem. Required conditions on spaces are also given, guaranteeing the existence and uniqueness of solutions. The fractional powers of the involved operator B X have been investigated in the space which consists of continuous functions u on [ 0 , ∞ ) without assuming u ( 0 ) = 0 . This enables us to refine some previous results and obtain the required abstract results when the operator B X is not necessarily densely defined.

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“…We quote papers [3,34], where for some δ ∈ (0, 1) and appropriate Banach spaces X, estimates like t −δ u X were obtained in the case of initial-boundary problems to autonomous linear subdiffusion equations. As for solvability of corresponding problems to equations like (1) in smooth classes: C δ ([0, T ], X), C k+α, k+α 2 ν (Ω T ), we refer to [10,13,16,22,27,23,24,31,40].…”

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confidence: 99%

Linear subdiffusion in weighted fractional Hölder spaces

Krasnoschok

Vasylyeva

2022

EECT

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For <inline-formula><tex-math id="M1">\begin{document}$ \nu\in(0,1) $\end{document}</tex-math></inline-formula>, we investigate the nonautonomous subdiffusion equation:<disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \mathbf{D}_{t}^{\nu}u-\mathcal{L}u = f(x,t), $\end{document} </tex-math></disp-formula>where <inline-formula><tex-math id="M2">\begin{document}$ \mathbf{D}_{t}^{\nu} $\end{document}</tex-math></inline-formula> is the Caputofractional derivative and <inline-formula><tex-math id="M3">\begin{document}$ \mathcal{L} $\end{document}</tex-math></inline-formula> is a uniformly ellipticoperator with smooth coefficients depending on time. Undersuitable conditions on the given data and a minimal number (i.e.the necessary number) of compatibility conditions, the globalclassical solvability to the related initial-boundary valueproblems are established in the weighted fractional Hölderspaces.

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On maximal regularity for the Cauchy-Dirichlet parabolic problem with fractional time derivative

Cited by 17 publications

References 31 publications

Direct and Inverse Fractional Abstract Cauchy Problems

Direct and Inverse Fractional Abstract Cauchy Problems

Fractional Cauchy Problems for Infinite Interval Case-II

Linear subdiffusion in weighted fractional Hölder spaces

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