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 Uspekhi Mat. Nauk, 1976, Volume 31, Issue 6(192), Pages 142–166 (Mi umn4013)

An analogue of Saint-Venant's principle and the uniqueness of solutions of boundary value problems for parabolic equations in unbounded domains

O. A. Oleinik, G. A. Iosif'yan

Abstract: Tikhonov's paper “A uniqueness theorem for the equation of heat conduction” [1], published in 1935, has had a great influence on the development of the theory of partial differential equations. In this paper he proved a uniqueness theorem for the solution of the Cauchy problem for the equation of heat conduction in certain classes of functions of exponential growth, and constructed examples of solutions to show non-uniqueness in wider classes of functions. Much research has been devoted to problems arising from Tikhonov's paper, and to the subsequent generalization and development of his results (see [2]–[10], and elsewhere); this research forms a significant contribution to the theory of partial differential equations.
Here we study the question of the uniqueness of the solution of the Cauchy problem, of boundary value problems, and of a problem without initial conditions. We also study the asymptotic properties of solutions of second order parabolic equations, by using a method based on the derivation of a priori estimates for the solutions that are similar to Saint-Venant's principle in the theory of elasticity [11]. Another new approach, which allows us to investigate these questions for general parabolic systems with general boundary conditions, and to obtain an analogue of Tikhonov's theorem, is given in [8]–[10]. It uses the analyticity of solutions of certain auxiliary parabolic systems with respect to an additional independent variable.

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English version:
Russian Mathematical Surveys, 1976, 31:6, 153–178

Bibliographic databases:

UDC: 517.9
MSC: 35A05, 35K05, 35K20, 35K50

Citation: O. A. Oleinik, G. A. Iosif'yan, “An analogue of Saint-Venant's principle and the uniqueness of solutions of boundary value problems for parabolic equations in unbounded domains”, Uspekhi Mat. Nauk, 31:6(192) (1976), 142–166; Russian Math. Surveys, 31:6 (1976), 153–178

Citation in format AMSBIB
\Bibitem{OleIos76} \by O.~A.~Oleinik, G.~A.~Iosif'yan \paper An analogue of Saint-Venant's principle and the uniqueness of solutions of boundary value problems for parabolic equations in unbounded domains \jour Uspekhi Mat. Nauk \yr 1976 \vol 31 \issue 6(192) \pages 142--166 \mathnet{http://mi.mathnet.ru/umn4013} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=481411} \zmath{https://zbmath.org/?q=an:0342.35026|0366.35046} \transl \jour Russian Math. Surveys \yr 1976 \vol 31 \issue 6 \pages 153--178 \crossref{https://doi.org/10.1070/RM1976v031n06ABEH001583} 

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Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. O. A. Oleinik, G. A. Iosif'yan, “Removable singularities on the boundary and uniqueness of solutions of boundary-value problems for second-order elliptic and parabolic equations”, Funct. Anal. Appl., 11:3 (1977), 206–217
2. O. A. Oleinik, E. V. Radkevich, “The method of introducing a parameter in the study of evolutionary equations”, Russian Math. Surveys, 33:5 (1978), 7–84
3. R. R. Kadyrov, “The asymptotic behaviour of solutions of boundary-value problems for a second-order parabolic equation as $t\to\infty$”, Russian Math. Surveys, 35:4 (1980), 169–170
4. G. I. Khil'kevich, “An analogue of Saint-Venant's principle, the Cauchy problem and the first boundary-value problem in an unbounded domain for pseudo-parabolic equations”, Russian Math. Surveys, 36:3 (1981), 252–253
5. V. A. Kondrat'ev, O. A. Oleinik, “Boundary-value problems for partial differential equations in non-smooth domains”, Russian Math. Surveys, 38:2 (1983), 1–66
6. O. A. Oleinik, “Examples of the non-uniqueness of the solution of the boundary-value problem for a parabolic equation in an unbounded domain”, Russian Math. Surveys, 38:1 (1983), 209–210
7. A. G. Gagnidze, “On uniqueness classes of the solutions of boundary-value problems for second-order parabolic equations in an unbounded domain”, Russian Math. Surveys, 39:6 (1984), 209–210
8. V. I. Arnol'd, M. I. Vishik, I. M. Gel'fand, Yu. V. Egorov, A. S. Kalashnikov, A. N. Kolmogorov, S. P. Novikov, S. L. Sobolev, “Ol'ga Arsen'evna Oleinik (on her sixtieth birthday)”, Russian Math. Surveys, 40:5 (1985), 267–287
9. F. Kh. Mukminov, “On uniform stabilization of solutions of the first mixed problem for a parabolic equation”, Math. USSR-Sb., 71:2 (1992), 331–353
10. N. M. Asadullin, F. Kh. Mukminov, “Uniqueness classes for a non-stationary system of Stokes equations in unbounded domains”, Sb. Math., 187:3 (1996), 315–333
11. A. E. Shishkov, “Propagation of perturbation in a singular Cauchy problem for degenerate quasilinear parabolic equations”, Sb. Math., 187:9 (1996), 1391–1410
12. A. E. Shishkov, A. G. Shchelkov, “Blow-up boundary regimes for general quasilinear parabolic equations in multidimensional domains”, Sb. Math., 190:3 (1999), 447–479
13. L. M. Kozhevnikova, F. Kh. Mukminov, “Estimates of the stabilization rate as $t\to\infty$ of solutions of the first mixed problem for a quasilinear system of second-order parabolic equations”, Sb. Math., 191:2 (2000), 235–273
14. L. M. Kozhevnikova, “On uniqueness classes of solutions of the first mixed problem for a quasi-linear second-order parabolic system in an unbounded domain”, Izv. Math., 65:3 (2001), 469–484
15. T. D. Dzhuraev, A. R. Khashimov, “O suschestvovanii reshenii pervoi kraevoi zadachi dlya uravnenii tretego poryadka sostavnogo tipa v neogranichennoi oblasti”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 19, SamGTU, Samara, 2003, 5–7
16. D. A. Sapronov, A. E. Shishkov, “Asymptotic behaviour of supports of solutions of quasilinear many-dimensionsal parabolic equations of non-stationary diffusion-convection type”, Sb. Math., 197:5 (2006), 753–790
17. N. M. Bokalo, “Correctness of the first boundary-value problem and the Cauchy problem for some quasilinear parabolic systems without conditions at infinity”, J. Math. Sci. (N. Y.), 135:1 (2006), 2625–2636
18. L. M. Kozhevnikova, “Uniqueness classes for solutions in unbounded domains of the first mixed problem for the equation $u_t=Au$ with quasi-elliptic operator $A$”, Sb. Math., 198:1 (2007), 55–96
19. B. Karp, D. Durban, “Saint-Venant’s Principle in Dynamics of Structures”, Appl. Mech. Rev, 64:2 (2011), 020801
20. V. F. Vil'danova, F. Kh. Mukminov, “Anisotropic uniqueness classes for a degenerate parabolic equation”, Sb. Math., 204:11 (2013), 1584–1597
21. V. F. Vil'danova, F. Kh. Mukminov, “Täcklind uniqueness classes for heat equation on noncompact Riemannian manifolds”, Ufa Math. J., 7:2 (2015), 55–63
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