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

This article is cited in 20 scientific papers (total in 21 papers)

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
Received: 30.08.1976

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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    Citing articles on Google Scholar: Russian citations, English citations
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    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  mathnet  crossref  mathscinet  zmath
    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  mathnet  crossref  mathscinet  zmath
    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  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    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  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    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  mathnet  crossref  mathscinet  zmath  isi
    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  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    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  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    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  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    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  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    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  mathnet  crossref  crossref  mathscinet  zmath  isi
    11. A. E. Shishkov, “Propagation of perturbation in a singular Cauchy problem for degenerate quasilinear parabolic equations”, Sb. Math., 187:9 (1996), 1391–1410  mathnet  crossref  crossref  mathscinet  zmath  isi
    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  mathnet  crossref  crossref  mathscinet  zmath  isi
    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  mathnet  crossref  crossref  mathscinet  zmath  isi
    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  mathnet  crossref  crossref  mathscinet  zmath
    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  mathnet  crossref  elib
    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  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    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  mathnet  crossref  mathscinet  zmath
    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  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    19. B. Karp, D. Durban, “Saint-Venant’s Principle in Dynamics of Structures”, Appl. Mech. Rev, 64:2 (2011), 020801  crossref
    20. V. F. Vil'danova, F. Kh. Mukminov, “Anisotropic uniqueness classes for a degenerate parabolic equation”, Sb. Math., 204:11 (2013), 1584–1597  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    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  mathnet  crossref  isi  elib
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