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.
Keywords and phrases
degenerate evolution equation differential equation of fractional order analytical in a sector resolving family of operators initial-boundary-value problem
AMS Subject Classification
34G10 34A08 35R11
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E. G. Bajlekova, Fractional evolution equations in Banach spaces, Ph.D. thesis, Eindhoven Univ. of Technology (2001).Google Scholar
P. Clément, H. J. A. M. Heijmans, S. Angenent, C. J. van Duijn, and B. de Pagter, One-Parameter Semigroups, North-Holland, Amsterdam (1987).zbMATHGoogle Scholar
V. E. Fedorov, “Degenerate strongly continuous semigroups of operator,” Algebra Anal., 12, No. 3, 173–200 (2000).MathSciNetGoogle Scholar
V. E. Fedorov, “Holomorphic resolving semigroups for equations of Sobolev Type in locally convex spaces,” Mat. Sb., 195, No. 8, 131–160 (2004).CrossRefGoogle Scholar
V. E. Fedorov and D. M. Gordievskikh, “Resolving operators of degenerate evolution equations with fractional time derivatives,” Izv. Vyssh. Ucheb. Zaved. Ser. Mat., 1, 71–83 (2015).zbMATHGoogle Scholar
V. E. Fedorov and D. M. Gordievskikh, “Solutions of initial-boundary-value problems for certain degenerate systems with fractional time derivatives,” Izv. Irkutsk. Univ. Ser. Mat., 12, 12–22 (2015).zbMATHGoogle Scholar
V. E. Fedorov, D. M. Gordievskikh, and M. V. Plekhanova, “Equation in Banach spaces with degenerate operators under the sign of fractional derivative,” Differ. Uravn., 51, No. 10, 1367–1375 (2015).zbMATHGoogle Scholar
V. E. Fedorov, R. R. Nazhimov, and D. M. Gordievskikh, “Initial-value problem for a class of fractional order inhomogeneous equations in Banach spaces,” AIP Conf. Proc., 1759, 020008 (2016).CrossRefGoogle Scholar
V. E. Fedorov, E. A. Romanova, and A. Debbouche, “Analytic in a sector resolving families of operators for degenerate evolution equations of a fractional order,” Sib. Zh. Chist. Prikl. Mat., 16, No. 2, 93–107 (2016).zbMATHGoogle Scholar
V. A. Kostin, “On the Solomyal–Yosida theorem for analytical semigroups,” Algebra Anal., 11, No. 1, 118–140 (1999).Google Scholar