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 Mat. Sb. (N.S.), 1975, Volume 97(139), Number 2(6), Pages 242–261 (Mi msb3650)

Some properties of a generalized solution of the second boundary-value problem for a parabolic equation

A. K. Gushchin

Abstract: We establish some properties (bounds in $L_p(\Omega)$ for $p\geqslant1$, absolute continuity of the entropy, etc.) for a solution in a cylindrical domain $\Omega\times\{t>0\}$, where $\Omega$ is an arbitrary, unbounded in general, domain of $R_n$, of the second boundary-value problem for a linear uniformly-parabolic equation of second order:
\begin{gather*} \frac{\partial u}{\partial t}=\sum_{i,j=1}^n\frac\partial{\partial x_i}(a_{ij}(t,x)\frac{\partial u(t,x)}{\partial x_j}),

Bibliography: 2 titles.

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English version:
Mathematics of the USSR-Sbornik, 1975, 26:2, 225–244

Bibliographic databases:

UDC: 517.947
MSC: 35K20, 35B45

Citation: A. K. Gushchin, “Some properties of a generalized solution of the second boundary-value problem for a parabolic equation”, Mat. Sb. (N.S.), 97(139):2(6) (1975), 242–261; Math. USSR-Sb., 26:2 (1975), 225–244

Citation in format AMSBIB
\Bibitem{Gus75} \by A.~K.~Gushchin \paper Some properties of a~generalized solution of the second boundary-value problem for a~parabolic equation \jour Mat. Sb. (N.S.) \yr 1975 \vol 97(139) \issue 2(6) \pages 242--261 \mathnet{http://mi.mathnet.ru/msb3650} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=422869} \zmath{https://zbmath.org/?q=an:0323.35041} \transl \jour Math. USSR-Sb. \yr 1975 \vol 26 \issue 2 \pages 225--244 \crossref{https://doi.org/10.1070/SM1975v026n02ABEH002478} 

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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. A. K. Gushchin, “On an estimate of the Dirichlet integral in unbounded domains”, Math. USSR-Sb., 28:2 (1976), 249–261
2. A. K. Gushchin, “Stabilization of the solutions of the second boundary value problem for a second order parabolic equation”, Math. USSR-Sb., 30:4 (1976), 403–440
3. A. K. Guščin, “On the behaviour ast→∞ of solutions of the second mixed problem for a second-order parabolic equation”, Appl Math Optim, 6:1 (1980), 169
4. V. I. Ushakov, “Stabilization of solutions of the third mixed problem for a second order parabolic equation in a noncylindrical domain”, Math. USSR-Sb., 39:1 (1981), 87–105
5. F. Kh. Mukminov, “Stabilization of solutions of the first mixed problem for a parabolic equation of second order”, Math. USSR-Sb., 39:4 (1981), 449–467
6. A. K. Gushchin, “On the uniform stabilization of solutions of the second mixed problem for a parabolic equation”, Math. USSR-Sb., 47:2 (1984), 439–498
7. A. I. Ibragimov, “Some qualitative properties of solutions of the mixed problem for equations of elliptic type”, Math. USSR-Sb., 50:1 (1985), 163–176
8. A. K. Gushchin, V. P. Mikhailov, Yu. A. Mikhailov, “On uniform stabilization of the solution of the second mixed problem for a second order parabolic equation”, Math. USSR-Sb., 56:1 (1987), 141–162
9. A. V. Lezhnev, “On the behavior, for large time values, of nonnegative solutions of the second mixed problem for a parabolic equation”, Math. USSR-Sb., 57:1 (1987), 195–209
10. Carl S., “The Efficiency of Monotone Iterative Technique in Semilinear Parabolic and Elliptic Edge-Value Problems”, Z. Angew. Math. Mech., 66:9 (1986), 429–434
11. Mukminov F., “Decrease with Time of the Norm of a Solution of a Mixed Problem for a High-Order Parabolic Equation”, Differ. Equ., 23:10 (1987), 1215–1220
12. Lezhnev A., “Bounds of Green-Function and Solutions of a 2nd Mixed Problem for a Parabolic Equation”, Differ. Equ., 25:4 (1989), 478–486
13. 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
14. Tedeev A., “Estimation of the Stabilization Rate for T -] Infinity of a Solution of a 2nd Mixed Problem for a 2nd-Order Quasi-Linear Parabolic Equation”, Differ. Equ., 27:10 (1991), 1274–1283
15. Mukminov F., “Decay of the Solution of the Mixed Problem for a Linearized System of Navier–Stokes Equations”, Differ. Equ., 28:8 (1992), 1156–1164
16. Mukminov F., “Decay Result for Solution of the Mixed Problem for the Navier–Stokes Equations in Domain with Incompact Boundary”, Dokl. Akad. Nauk, 324:6 (1992), 1149–1151
17. F. Kh. Mukminov, “On uniform stabilization of solutions of the exterior problem for the Navier–Stokes equations”, Russian Acad. Sci. Sb. Math., 81:2 (1995), 297–320
18. Rozkosz A., “On a Decomposition of Symmetric Diffusions with Reflecting Boundary Conditions”, Stoch. Process. Their Appl., 103:1 (2003), 101–122
19. F. Kh. Mukminov, I. M. Bikkulov, “Stabilization of the norm of the solution of a mixed problem in an unbounded domain for parabolic equations of orders 4 and 6”, Sb. Math., 195:3 (2004), 413–440
20. V. F. Gilimshina, F. Kh. Mukminov, “Ob ubyvanii resheniya vyrozhdayuschegosya lineinogo parabolicheskogo uravneniya”, Ufimsk. matem. zhurn., 3:4 (2011), 43–56
21. V. F. Vil'danova, “On decay of solution to linear parabolic equation with double degeneracy”, Ufa Math. J., 8:1 (2016), 35–50
22. V. F. Vildanova, “Uravnenie agregatsii s anizotropnoi diffuziei”, Tr. IMM UrO RAN, 23, no. 3, 2017, 58–73
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