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Mat. Sb., 2004, Volume 195, Number 8, Pages 131–160 (Mi msb841)  

This article is cited in 13 scientific papers (total in 13 papers)

Holomorphic solution semigroups for Sobolev-type equations in locally convex spaces

V. E. Fedorov

Chelyabinsk State University

Abstract: The problem of the existence of an exponentially bounded solution semigroup strongly holomorphic in a sector is studied for a Sobolev-type linear equation
\begin{equation} L\dot u=Mu \end{equation}
with continuous operator $L\colon\mathfrak U\to\mathfrak F$, $\ker L\ne\{0\}$, and closed densely defined operator $M\colon\operatorname{dom}M\to\mathfrak F$, where $\mathfrak U$ and $\mathfrak F$ are sequentially complete locally convex spaces. It is shown that the condition of the $(L,p)$-sectoriality of the operator $M$, which generalizes the well known condition of sectoriality, is necessary and sufficient for the existence of such semigroups degenerate at the $M$-associated vectors of the operator $L$ of height $p$ and lower and the existence of pairs of invariant subspaces of the operators $L$ and $M$. Generalizations of Yosida's theorem and results on the existence of a holomorphic solution semigroup for equation (1) in Banach spaces are obtained. These results are used in the study of the weakened Cauchy problem for equation (1) and for the corresponding non-linear equation. One application of the abstract results is a theorem on sufficient conditions for the solubility of the Cauchy problem for a class of equations in Fréchet spaces of a special kind. It is used in the analysis of the periodic Cauchy problem for a partial differential equation with displacement not solved with respect to the time derivative.

DOI: https://doi.org/10.4213/sm841

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English version:
Sbornik: Mathematics, 2004, 195:8, 1205–1234

Bibliographic databases:

UDC: 517.9
MSC: Primary 47D06; Secondary 47N20
Received: 25.07.2003 and 30.01.2004

Citation: V. E. Fedorov, “Holomorphic solution semigroups for Sobolev-type equations in locally convex spaces”, Mat. Sb., 195:8 (2004), 131–160; Sb. Math., 195:8 (2004), 1205–1234

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. E. Fedorov, O. A. Ruzakova, “On solvability of perturbed Sobolev type equations”, St. Petersburg Math. J., 20:4 (2009), 645–664  mathnet  crossref  mathscinet  zmath  isi
    2. V. E. Fedorov, “Golomorfnye polugruppy operatorov s silnym vyrozhdeniem”, Vestnik ChelGU, 2008, no. 10, 68–74  mathnet
    3. A. V. Urazaeva, V. E. Fedorov, “On the Well-Posedness of the Prediction-Control Problem for Certain Systems of Equations”, Math. Notes, 85:3 (2009), 426–436  mathnet  crossref  crossref  mathscinet  zmath  isi
    4. M. V. Plekhanova, A. F. Islamova, “Issledovanie linearizovannoi sistemy uravnenii Bussineska metodami teorii vyrozhdennykh polugrupp”, Vestnik ChelGU, 2009, no. 11, 62–69  mathnet
    5. O. A. Ruzakova, E. A. Oleinik, “Ob upravlyaemosti lineinykh uravnenii sobolevskogo tipa s otnositelno sektorialnym operatorom”, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 2012, no. 11, 54–61  mathnet
    6. Fedorov V.E., Debbouche A., “A Class of Degenerate Fractional Evolution Systems in Banach Spaces”, Differ. Equ., 49:12 (2013), 1569–1576  crossref  mathscinet  zmath  isi
    7. M. V. Plekhanova, “Sistemy optimalnosti dlya vyrozhdennykh raspredelennykh zadach upravleniya”, Vestnik ChelGU, 2013, no. 16, 60–70  mathnet  mathscinet  elib
    8. A. V. Keller, Dzh. K. Al-Delfi, “Golomorfnye vyrozhdennye gruppy operatorov v kvazibanakhovykh prostranstvakh”, Vestn. Yuzhno-Ur. un-ta. Ser. Matem. Mekh. Fiz., 7:1 (2015), 20–27  mathnet  elib
    9. A. L. Shestakov, G. A. Sviridyuk, M. D. Butakova, “The mathematical modelling of the production of construction mixtures with prescribed properties”, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 8:1 (2015), 100–110  mathnet  crossref  elib
    10. A. A. Zamyshlyaeva, D. K. T. Al-Isavi, “Golomorfnye vyrozhdennye polugruppy operatorov i evolyutsionnye uravneniya sobolevskogo tipa v kvazisobolevykh prostranstvakh posledovatelnostei”, Vestn. Yuzhno-Ur. un-ta. Ser. Matem. Mekh. Fiz., 7:4 (2015), 27–36  mathnet  crossref  elib
    11. Fedorov V.E., Gordievskikh D.M., Plekhanova M.V., “Equations in Banach Spaces With a Degenerate Operator Under a Fractional Derivative”, Differ. Equ., 51:10 (2015), 1360–1368  crossref  mathscinet  zmath  isi  elib
    12. J. K. T. Al-Isawi, “On kernels and images of resolving analytic degenerate semigroups in quasi-Sobolev spaces”, J. Comp. Eng. Math., 3:1 (2016), 10–19  mathnet  crossref  mathscinet  zmath  elib
    13. V. E. Fedorov, E. A. Romanova, A. Debbouche, “Analytic in a sector resolving families of operators for degenerate evolution equations of a fractional order”, J. Math. Sci., 228:4 (2018), 380–394  mathnet  crossref  crossref
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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