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**On analytical in a sector resolving families of operators for strongly degenerate evolution equations of higher and fractional orders.**
*(English.
Russian original)*
Zbl 1417.34010

J. Math. Sci., New York 236, No. 6, 663-678 (2019); translation from Itogi Nauki Tekh., Ser. Sovrem. Mat. Prilozh., Temat. Obz. 137, 82-96 (2017).

Summary: In this paper, we study a class of linear evolution equations of fractional order that are degenerate on the kernel of the operator under the sign of the derivative and on its relatively generalized eigenvectors. We prove that in the case considered, in contrast to the case of first-order degenerate equations and equations of fractional order with weak degeneration (i.e., degeneration only on the kernel of the operator under the sign of the derivative), the family of analytical in a sector operators does not vanish on relative generalized eigenspaces of the operator under the sign of the derivative, has a singularity at zero, and hence does not determine any solution of a strongly degenerate equation of fractional order. For the case of a strongly degenerate equation of integer order this fact does not hold, but the behavior of the family of resolving operators at zero cannot be examined by ordinary method.

### MSC:

34A08 | Fractional ordinary differential equations |

34G10 | Linear differential equations in abstract spaces |

35R11 | Fractional partial differential equations |

### Keywords:

degenerate evolution equation; differential equation of fractional order; analytical in a sector resolving family of operators; initial-boundary-value problem
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\textit{V. E. Fedorov} and \textit{E. A. Romanova}, J. Math. Sci., New York 236, No. 6, 663--678 (2019; Zbl 1417.34010); translation from Itogi Nauki Tekh., Ser. Sovrem. Mat. Prilozh., Temat. Obz. 137, 82--96 (2017)

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### References:

[1] | E. G. Bajlekova, Fractional evolution equations in Banach spaces, Ph.D. thesis, Eindhoven Univ. of Technology (2001). · Zbl 0989.34002 |

[2] | P. Clément, H. J. A. M. Heijmans, S. Angenent, C. J. van Duijn, and B. de Pagter, One-Parameter Semigroups, North-Holland, Amsterdam (1987). · Zbl 0636.47051 |

[3] | Fedorov, VE, Degenerate strongly continuous semigroups of operator, Algebra Anal., 12, 173-200, (2000) |

[4] | Fedorov, VE, Holomorphic resolving semigroups for equations of Sobolev Type in locally convex spaces, Mat. Sb., 195, 131-160, (2004) |

[5] | Fedorov, VE; Gordievskikh, DM, Resolving operators of degenerate evolution equations with fractional time derivatives, Izv. Vyssh. Ucheb. Zaved. Ser. Mat., 1, 71-83, (2015) · Zbl 1323.47048 |

[6] | Fedorov, VE; Gordievskikh, DM, Solutions of initial-boundary-value problems for certain degenerate systems with fractional time derivatives, Izv. Irkutsk. Univ. Ser. Mat., 12, 12-22, (2015) · Zbl 1344.47054 |

[7] | Fedorov, VE; Gordievskikh, DM; Plekhanova, MV, Equation in Banach spaces with degenerate operators under the sign of fractional derivative, Differ. Uravn., 51, 1367-1375, (2015) · Zbl 1330.47095 |

[8] | Fedorov, VE; Nazhimov, RR; Gordievskikh, DM, Initial-value problem for a class of fractional order inhomogeneous equations in Banach spaces, AIP Conf. Proc., 1759, 020008, (2016) |

[9] | Fedorov, VE; Romanova, EA; Debbouche, A., Analytic in a sector resolving families of operators for degenerate evolution equations of a fractional order, Sib. Zh. Chist. Prikl. Mat., 16, 93-107, (2016) · Zbl 1399.34012 |

[10] | Kostin, VA, On the Solomyal-Yosida theorem for analytical semigroups, Algebra Anal., 11, 118-140, (1999) |

[11] | J. Prüss, Evolutionary Integral Equations and Applications, Springer, Basel (1993). |

[12] | Solomyak, MZ, Application of the theory of semigroups to the study of differential equations in Banach spaces, Dokl. Akad. Nauk SSSR, 122, 766-769, (1958) · Zbl 0090.09902 |

[13] | Sviridyuk, GA; Fedorov, VE, On units of analytical semigroups of operators with kernels, Sib. Mat. Zh., 39, 604-616, (1998) · Zbl 0908.47036 |

[14] | Yagi, A., Generation theorem of semigroup for multivalued linear operators, Osaka J. Math., 28, 385-410, (1991) · Zbl 0812.47045 |

[15] | K. Yosida, Functional Analysis, Springer-Verlag, Berlin-Göttingen-Heidelberg (1965). · Zbl 0126.11504 |

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