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Vestnik YuUrGU. Ser. Mat. Model. Progr., 2015, Volume 8, Issue 4, Pages 113–119 (Mi vyuru293)  

This article is cited in 7 scientific papers (total in 7 papers)

Short Notes

On some properties of solutions to one class of evolution Sobolev type mathematical models in quasi-Sobolev spaces

A. A. Zamyshlyaeva, D. K. T. Al-Isawi

South Ural State University, Chelyabinsk, Russian Federation

Abstract: Interest in Sobolev type equations has recently increased significantly, moreover, there arose a necessity for their consideration in quasi-Banach spaces. The need is dictated not so much by the desire to fill up the theory but by the aspiration to comprehend non-classical models of mathematical physics in quasi-Banach spaces. Notice that the Sobolev type equations are called evolutionary if solutions exist only on ${{\mathbb R}}_{{\mathbf +}}$.
The theory of holomorphic degenerate semigroups of operators constructed earlier in Banach spaces and Frechet spaces is transferred to quasi-Sobolev spaces of sequences. This article contains results about existence of the exponential dichotomies of solutions to evolution Sobolev type equation in quasi-Sobolev spaces. To obtain this result we proved the relatively spectral theorem and the existence of invariant spaces of solutions.
The article besides the introduction and references contains two paragraphs. In the first one, quasi-Banach spaces, quasi-Sobolev spaces and polynomials of Laplace quasi-operator are defined. Moreover the conditions for existence of degenerate holomorphic operator semigroups in quasi-Banach spaces of sequences are obtained. In other words, we prove the first part of the generalization of the Solomyak–Iosida theorem to quasi-Banach spaces of sequences. In the second paragraph the phase space of the homogeneous equation is constructed. Here we show the existence of invariant spaces of equation and get the conditions for exponential dichotomies of solutions.

Keywords: holomorphic degenerate semigroups; quasi-Banach spaces; quasi-Sobolev spaces; invariant space; exponential dichotomy of solution.

DOI: https://doi.org/10.14529/mmp150410

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UDC: 517.9
MSC: 46A16, 47D03, 34D09
Received: 21.09.2015
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Citation: A. A. Zamyshlyaeva, D. K. T. Al-Isawi, “On some properties of solutions to one class of evolution Sobolev type mathematical models in quasi-Sobolev spaces”, Vestnik YuUrGU. Ser. Mat. Model. Progr., 8:4 (2015), 113–119

Citation in format AMSBIB
\Bibitem{ZamAl-15}
\by A.~A.~Zamyshlyaeva, D.~K.~T.~Al-Isawi
\paper On some properties of solutions to one class of evolution Sobolev type mathematical models in quasi-Sobolev spaces
\jour Vestnik YuUrGU. Ser. Mat. Model. Progr.
\yr 2015
\vol 8
\issue 4
\pages 113--119
\mathnet{http://mi.mathnet.ru/vyuru293}
\crossref{https://doi.org/10.14529/mmp150410}
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\elib{http://elibrary.ru/item.asp?id=24989387}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. J. K. T. Al-Isawi, “On some properties of solutions to Dzektser mathematical model in quasi-Sobolev spaces”, J. Comp. Eng. Math., 2:4 (2015), 27–36  mathnet  crossref  elib
    2. M. A. Sagadeeva, “Mathematical bases of optimal measurements theory in nonstationary case”, J. Comp. Eng. Math., 3:3 (2016), 19–32  mathnet  crossref  mathscinet  elib
    3. J. K. T. Al-Isawi, A. A. Zamyshlyaeva, “Computational experiment for one class of evolution mathematical models in quasi-Sobolev spaces”, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 9:4 (2016), 141–147  mathnet  crossref  elib
    4. M. A. Sagadeeva, “Vyrozhdennye potoki razreshayuschikh operatorov dlya nestatsionarnykh uravnenii sobolevskogo tipa”, Vestn. Yuzhno-Ur. un-ta. Ser. Matem. Mekh. Fiz., 9:1 (2017), 22–30  mathnet  crossref  elib
    5. J. K. T. Al-Isawi, “Computational experiments for one class of mathematical models in thermodynamics and hydrodynamics”, J. Comp. Eng. Math., 4:1 (2017), 16–26  mathnet  crossref  elib
    6. I. S. Strepetova, L. M. Fatkullina, G. A. Zakirova, “Spectral problems for one mathematical model of hydrodynamics”, J. Comp. Eng. Math., 4:1 (2017), 48–56  mathnet  crossref  mathscinet  elib
    7. D. E. Shafranov, N. V. Adukova, “Solvability of the Showalter–Sidorov problem for Sobolev type equations with operators in the form of first-order polynomials from the Laplace–Beltrami operator on differential forms”, J. Comp. Eng. Math., 4:3 (2017), 27–34  mathnet  crossref  mathscinet  elib
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