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Uspekhi Mat. Nauk, 1968, Volume 23, Issue 1(139), Pages 45–90 (Mi umn5592)  

This article is cited in 29 scientific papers (total in 29 papers)

Quasilinear elliptic and parabolic equations of arbitrary order

Yu. A. Dubinskii


Abstract: This paper is a survey of recent results on the solution of boundary value problems for quasilinear elliptic and parabolic equations of order $2m$, of divergent form. The main results in this direction were first obtained in 1961 by Vishik, Browder, the author and others, and are presented in the first part of the paper. We also indicate the spaces in which the elliptic and parabolic operators induce homeomorphisms in the strongly elliptic case. When the variation of the operator is merely semibounded below, the Dirichlet problem is soluble for any right-hand side, though not uniquely. In the second part we present the work of several authors concerning the solution of operator equations in Banach spaces, among them Minty, Browder, Leray, Lions, Dubinskii, Pokhozhaev. The results are then applied to non-linear differential equations.

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English version:
Russian Mathematical Surveys, 1968, 23:1, 45–91

Bibliographic databases:

UDC: 517.9
MSC: 35H30, 35K60, 35J65, 40A05, 47B48
Received: 03.05.1966

Citation: Yu. A. Dubinskii, “Quasilinear elliptic and parabolic equations of arbitrary order”, Uspekhi Mat. Nauk, 23:1(139) (1968), 45–90; Russian Math. Surveys, 23:1 (1968), 45–91

Citation in format AMSBIB
\Bibitem{Dub68}
\by Yu.~A.~Dubinskii
\paper Quasilinear elliptic and parabolic equations of arbitrary order
\jour Uspekhi Mat. Nauk
\yr 1968
\vol 23
\issue 1(139)
\pages 45--90
\mathnet{http://mi.mathnet.ru/umn5592}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=228826}
\zmath{https://zbmath.org/?q=an:0157.18101|0162.42303}
\transl
\jour Russian Math. Surveys
\yr 1968
\vol 23
\issue 1
\pages 45--91
\crossref{https://doi.org/10.1070/RM1968v023n01ABEH001233}


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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. L. R. Volevich, S. G. Gindikin, “The Cauchy problem for pluriparabolic differential equations. II”, Math. USSR-Sb., 7:2 (1969), 205–226  mathnet  crossref  zmath
    2. P. P. Mosolov, “Variational methods in nonstationary problems (parabolic case)”, Math. USSR-Izv., 4:2 (1970), 431–463  mathnet  crossref  mathscinet  zmath
    3. P. D. Mil'man, “Fixed points and the degree of a mapping”, Math. USSR-Sb., 16:2 (1972), 287–306  mathnet  crossref  mathscinet  zmath
    4. Yu. A. Dubinskii, “On some noncoercive nonlinear equations”, Math. USSR-Sb., 16:3 (1972), 323–332  mathnet  crossref  mathscinet  zmath
    5. V. V. Zhikov, “Monotonicity in the theory of almost periodic solutions of nonlinear operator equations”, Math. USSR-Sb., 19:2 (1973), 209–223  mathnet  crossref  mathscinet  zmath
    6. 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”, Math. USSR-Sb., 27:2 (1975), 143–162  mathnet  crossref  mathscinet  zmath
    7. R. C. MacCamy, “Stability Theorems for a Class of Functional Differential Equations”, SIAM J Appl Math, 30:3 (1976), 557  crossref  mathscinet  zmath
    8. R.C MacCamy, “Remarks on frequency domain methods for Volterra integral equations”, Journal of Mathematical Analysis and Applications, 55:3 (1976), 555  crossref
    9. V. V. Zhikov, B. M. Levitan, “Favard theory”, Russian Math. Surveys, 32:1 (1977), 129–180  mathnet  crossref  mathscinet  zmath
    10. Mitsuhiro Nakao, “Decay of solutions of some nonlinear evolution equations”, Journal of Mathematical Analysis and Applications, 60:2 (1977), 542  crossref
    11. A. I. Koshelev, “Regularity of solutions of quasi-linear elliptic systems”, Russian Math. Surveys, 33:4 (1978), 1–52  mathnet  crossref  mathscinet  zmath
    12. J.Tinsley Oden, “Existence theorems for a class of problems in nonlinear elasticity”, Journal of Mathematical Analysis and Applications, 69:1 (1979), 51  crossref
    13. G. G. Kazaryan, G. A. Karapetyan, “On the convergence of Galerkin approximations to the solution of the Dirichlet problem for some general equations”, Math. USSR-Sb., 52:2 (1985), 285–299  mathnet  crossref  mathscinet  zmath
    14. M. V. Vladimirov, “Solvability of a mixed problem for the nonlinear Schrödinger equation”, Math. USSR-Sb., 58:2 (1987), 525–540  mathnet  crossref  mathscinet  zmath
    15. Yu. A. Dubinskii, “Analytic “boundary-value” problems on the plane”, Russian Math. Surveys, 52:3 (1997), 501–550  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    16. V. V. Zhikov, M. E. Rychago, “Homogenization of non-linear second-order elliptic equations in perforated domains”, Izv. Math., 61:1 (1997), 69–88  mathnet  crossref  crossref  mathscinet  zmath  isi
    17. K. O. Besov, “Norm-Generating Pseudodifferential Operators in the Spaces $W_p^s(\mathbb R^n)$”, Proc. Steklov Inst. Math., 232 (2001), 52–65  mathnet  mathscinet  zmath
    18. P. V. Vinogradova, A. G. Zarubin, “O metode Galerkina dlya kvazilineinykh parabolicheskikh uravnenii v netsilindricheskoi oblasti”, Dalnevost. matem. zhurn., 3:1 (2002), 3–17  mathnet
    19. M. M. Karchevskii, A. E. Fedotov, “Smeshannyi metod konechnykh elementov dlya kvazilineinykh vyrozhdayuschikhsya ellipticheskikh uravnenii”, Uchen. zap. Kazan. gos. un-ta. Ser. Fiz.-matem. nauki, 147, no. 3, Izd-vo Kazanskogo un-ta, Kazan, 2005, 127–140  mathnet  zmath
    20. A. A. Mamikonyan, “Nachalno-kraevaya zadacha dlya odnogo klassa nelineinykh uravnenii tipa Soboleva”, Uch. zapiski EGU, ser. Fizika i Matematika, 2006, no. 2, 33–40  mathnet
    21. V. K. Bulgakov, G. L. Shatov, “Optimal control in a macroeconomic problem”, Comput. Math. Math. Phys., 47:8 (2007), 1253–1267  mathnet  crossref  mathscinet  elib  elib
    22. Sh. M. Nasibov, “On an Integral Inequality and Its Application to the Proof of the Entropy Inequality”, Math. Notes, 84:2 (2008), 218–223  mathnet  crossref  crossref  mathscinet  isi
    23. Sh. M. Nasibov, “A sharp constant in a Sobolev–Nirenberg inequality and its application to the Schrödinger equation”, Izv. Math., 73:3 (2009), 555–577  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    24. V. K. Bulgakov, V. V. Strigunov, “Optimal control and optimal trajectories of regional macroeconomic dynamics based on the Pontryagin maximum principle”, Comput. Math. Math. Phys., 49:5 (2009), 748–761  mathnet  crossref  zmath  isi  elib  elib
    25. M. A. Nurmamedov, “O razreshimosti kraevykh zadach dlya kvazilineinoi sistemy uravnenii smeshanno-sostavnogo tipa s menyayuschimsya napravleniem vremeni v mnogomernoi oblasti”, Vladikavk. matem. zhurn., 12:2 (2010), 46–61  mathnet  elib
    26. S. N. Timergaliev, I. R. Mavleev, “Solvability of the boundary value problem for a partial quasilinear differential equation of the fourth order”, Russian Math. (Iz. VUZ), 54:12 (2010), 45–50  mathnet  crossref  mathscinet  elib
    27. O. V. Glazyrina, M. F. Pavlova, “Issledovanie skhodimosti yavnoi raznostnoi skhemy dlya parabolicheskogo uravneniya s nelineinym nelokalnym prostranstvennym operatorom”, Uchen. zap. Kazan. un-ta. Ser. Fiz.-matem. nauki, 155, no. 4, Izd-vo Kazanskogo un-ta, Kazan, 2013, 24–39  mathnet
    28. A. G. Podgaev, N. E. Istomina, “O metodakh Faedo – Galerkina i monotonnosti v netsilindricheskoi oblasti dlya vyrozhdayuschegosya kvazilineinogo uravneniya”, Dalnevost. matem. zhurn., 14:1 (2014), 73–89  mathnet
    29. I. E. Egorov, E. S. Efimova, “A modified Galerkin method for semilinear parabolic equation with changing time direction”, J. Math. Sci., 228:4 (2018), 372–379  mathnet  crossref  crossref
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