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 Mat. Sb. (N.S.), 1975, Volume 98(140), Number 2(10), Pages 163–184 (Mi msb3704)

Sobolev spaces of infinite order and the behavior of solutions of some boundary value problems with unbounded increase of the order of the equation

Yu. A. Dubinskii

Abstract: In the study of the Cauchy–Dirichlet problem
\begin{gather} L(u)\equiv\sum_{|\alpha|=0}^\infty(-1)^{|\alpha|}D^\alpha A_\alpha(x, D^\gamma u)=h(x),\qquad x\in G,
D^\omega u\mid_{\partial G}=0,\qquad |\omega|=0,1,…, \end{gather}
infinite order Sobolev spaces
$$\overset\circ W ^\infty\{a_\alpha, p_\alpha\}\equiv\{u(x)\in C^\infty_0(G):\rho(u)\equiv\sum^\infty_{|\alpha|=0}a_\alpha\|D^\alpha u\|_{p_\alpha}^{p_\alpha}<\infty\},$$
naturally arise, where $a_\alpha\geqslant0$ and $p_\alpha\geqslant1$ are numerical sequences. In this paper criteria for the nontriviality of $\overset\circ W ^\infty\{a_\alpha,p_\alpha\}$ are established and the problem (1), (2) is investigated. Further, a theorem is obtained on the existence of the limit (as $m\to\infty$) of solutions of nonlinear $2m$th order boundary value problems of elliptic and hyperbolic type, from which, in particular, follows the solvability of the mixed problem for the nonlinear hyperbolic equation $u"+L(u)=h(t,x)$, $t\in[0,T]$, where $T>0$ is arbitrary.
Bibliography: 9 titles.

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English version:
Mathematics of the USSR-Sbornik, 1975, 27:2, 143–162

Bibliographic databases:

UDC: 517.946.9
MSC: Primary 46E35, 35J60, 35L35; Secondary 28A93

Citation: Yu. A. Dubinskii, “Sobolev spaces of infinite order and the behavior of solutions of some boundary value problems with unbounded increase of the order of the equation”, Mat. Sb. (N.S.), 98(140):2(10) (1975), 163–184; Math. USSR-Sb., 27:2 (1975), 143–162

Citation in format AMSBIB
\Bibitem{Dub75} \by Yu.~A.~Dubinskii \paper Sobolev spaces of infinite order and the behavior of solutions of some boundary value problems with unbounded increase of the order of the equation \jour Mat. Sb. (N.S.) \yr 1975 \vol 98(140) \issue 2(10) \pages 163--184 \mathnet{http://mi.mathnet.ru/msb3704} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=412580} \zmath{https://zbmath.org/?q=an:0324.46037} \transl \jour Math. USSR-Sb. \yr 1975 \vol 27 \issue 2 \pages 143--162 \crossref{https://doi.org/10.1070/SM1975v027n02ABEH002506} 

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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. Yu. A. Dubinskii, “Nontriviality of Sobolev spaces of infinite order for a full Euclidean space and a torus”, Math. USSR-Sb., 29:3 (1976), 393–401
2. Yu. A. Dubinskii, “Traces of functions from Sobolev spaces of infinite order and inhomogeneous problems for nonlinear equations”, Math. USSR-Sb., 34:5 (1978), 627–644
3. Yu. A. Dubinskii, “Limits of Banach spaces. Imbedding theorems. Applications to Sobolev spaces of infinite order”, Math. USSR-Sb., 38:3 (1981), 395–405
4. Tran Duc Van, “Elliptic equations of infinite order with arbitrary nonlinearities and corresponding function spaces”, Math. USSR-Sb., 41:2 (1982), 203–216
5. Van C., “Solvability of Boundary-Value-Problems for Degenerate Non-Linear Differential-Equations of Infinite-Order”, Differ. Equ., 16:10 (1980), 1202–1211
6. Dubinskii I., “A Method of Solving Partial-Differential Equations”, 258, no. 4, 1981, 780–784
7. Balashova G., “Behavior of Solutions of Certain Boundary-Value-Problems When the Order of the Equation Increases Indefinitely”, Differ. Equ., 17:2 (1981), 175–185
8. G. S. Balashova, “On extension theorems in spaces of infinitely differentiable functions”, Math. USSR-Sb., 46:3 (1983), 375–389
9. Kobilov A., “Non-Triviality of Some Spaces of Infinitely Differentiable Functions in Corner Domains and the Solvability of Non-Linear Elliptic-Equations”, 266, no. 5, 1982, 1040–1044
10. Kuchminskaya L., “Solvability of Mixed Problems for a Certain Class of Nonlinear Infinite-Order Differential-Equations”, no. 5, 1984, 8–11
11. Radyno Y., “Differential-Equations in a Banach-Space Scale”, Differ. Equ., 21:8 (1985), 971–979
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15. Yu. A. Dubinskii, “Sobolev spaces of infinite order”, Russian Math. Surveys, 46:6 (1991), 107–147
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21. M. Chrif, S. El Manouni, “Anisotropic equations in weighted Sobolev spaces of higher order”, Ricerche mat, 2009
22. Mostafa Bendahmane, Moussa Chrif, Said El Manouni, “Elliptic equations in weighted Sobolev spaces of infinite order with L <sup>1</sup> data”, Math Meth Appl Sci, 2009, n/a
23. Kowalewski A., “Optimal Control via Initial Conditions of Infinite Order Hyperbolic Systems”, 2012 17th International Conference on Methods and Models in Automation and Robotics (Mmar), IEEE, 2012, 212–215
24. B.G.aber Mohamed, “Boundary Control Problem of Infinite Order Distributed Hyperbolic Systems Involving Time Lags”, ICA, 03:03 (2012), 211
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26. G.M. Bahaa, S.A.A. El-Marouf, “Pareto Optimal Control For Mixed Neumann Infinite-Order Parabolic System With State-Control Constraints”, Journal of Taibah University for Science, 2014
27. M.H.ousseine Abdou, Moussa Chrif, Said El Manouni, “Parabolic Equations of Infinite Order withL1Data”, Abstract and Applied Analysis, 2014 (2014), 1
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