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 Mat. Sb., 2016, Volume 207, Number 1, Pages 3–44 (Mi msb8484)

Existence and qualitative properties of a solution of the first mixed problem for a parabolic equation with non-power-law double nonlinearity

È. R. Andriyanovaa, F. Kh. Mukminovb

a Ufa State Aviation Technical University
b Institute of Mathematics with Computing Centre, Ufa Science Centre, Russian Academy of Sciences, Ufa

Abstract: The first mixed problem is investigated for a certain class of parabolic equations with double non-power-law nonlinearities in a cylindrical domain of the form $D=(t>0)\times\Omega$. The domain $\Omega\subset \mathbb R^n$ can be unbounded. The existence of strong solutions in a Sobolev-Orlicz space is proved by the method of Galerkin approximations. A maximum principle is established, and upper and lower bounds characterizing the power-law decay of solution as $t\to \infty$ are proved. The uniqueness of the solution is proved under certain additional assumptions.
Bibliography: 29 titles.

Keywords: parabolic equation with double nonlinearity, $N$-functions, existence of a solution, estimate for the decay rate of a solution.

 Funding Agency Grant Number Russian Foundation for Basic Research 15-01-07920-a This work was supported by the Russian Foundation for Basic Research (grant no. 15-01-07920-a).

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DOI: https://doi.org/10.4213/sm8484

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English version:
Sbornik: Mathematics, 2016, 207:1, 1–40

Bibliographic databases:

UDC: 517.954+517.956.45+517.958:531.72
MSC: Primary 35K20; Secondary 35K55

Citation: È. R. Andriyanova, F. Kh. Mukminov, “Existence and qualitative properties of a solution of the first mixed problem for a parabolic equation with non-power-law double nonlinearity”, Mat. Sb., 207:1 (2016), 3–44; Sb. Math., 207:1 (2016), 1–40

Citation in format AMSBIB
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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. F. Kh. Mukminov, “Uniqueness of the renormalized solution of an elliptic-parabolic problem in anisotropic Sobolev-Orlicz spaces”, Sb. Math., 208:8 (2017), 1187–1206
2. V. F. Vildanova, “Uravnenie agregatsii s anizotropnoi diffuziei”, Tr. IMM UrO RAN, 23, no. 3, 2017, 58–73
3. V. F. Vil'danova, “Existence and uniqueness of a weak solution of a nonlocal aggregation equation with degenerate diffusion of general form”, Sb. Math., 209:2 (2018), 206–221
4. A. K. Gushchin, “The Luzin area integral and the nontangential maximal function for solutions to a second-order elliptic equation”, Sb. Math., 209:6 (2018), 823–839
5. A. K. Gushchin, “A criterion for the existence of $L_p$ boundary values of solutions to an elliptic equation”, Proc. Steklov Inst. Math., 301 (2018), 44–64
6. A. K. Gushchin, “On the Existence of $L_2$ Boundary Values of Solutions to an Elliptic Equation”, Proc. Steklov Inst. Math., 306 (2019), 47–65
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