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Fundam. Prikl. Mat., 2006, Volume 12, Issue 4, Pages 3–19 (Mi fpm956)  

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

Existence and uniqueness of solutions of degenerate parabolic equations with variable exponents of nonlinearity

S. N. Antontseva, S. I. Shmarevb

a University of Beira Interior
b Universidad de Oviedo

Abstract: We prove the existence and uniqueness of weak solutions of the Dirichlet problem for the nonlinear degenerate parabolic equations
$$ u_{t}=\operatorname{div}(a|u|^{\gamma(x,t)}\nabla u)+\mathbf{b}|u|^{\gamma(x,t)/2}\nabla u-c|u|^{\sigma (x,t)-2}u+d, $$
where $a$, $\mathbf{b}$, $c$, and $d$ are given functions of the arguments $x$, $t$, and $u(x,t)$, and the exponents of nonlinearity $\gamma(x,t)$ and $\sigma(x,t)$ are known measurable and bounded functions of their arguments.

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English version:
Journal of Mathematical Sciences (New York), 2008, 150:5, 2289–2301

Bibliographic databases:

UDC: 517.957+517.956.4

Citation: S. N. Antontsev, S. I. Shmarev, “Existence and uniqueness of solutions of degenerate parabolic equations with variable exponents of nonlinearity”, Fundam. Prikl. Mat., 12:4 (2006), 3–19; J. Math. Sci., 150:5 (2008), 2289–2301

Citation in format AMSBIB
\Bibitem{AntShm06}
\by S.~N.~Antontsev, S.~I.~Shmarev
\paper Existence and uniqueness of solutions of degenerate parabolic equations with variable exponents of nonlinearity
\jour Fundam. Prikl. Mat.
\yr 2006
\vol 12
\issue 4
\pages 3--19
\mathnet{http://mi.mathnet.ru/fpm956}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2314142}
\zmath{https://zbmath.org/?q=an:1151.35382}
\elib{http://elibrary.ru/item.asp?id=11143772}
\transl
\jour J. Math. Sci.
\yr 2008
\vol 150
\issue 5
\pages 2289--2301
\crossref{https://doi.org/10.1007/s10958-008-0129-6}
\elib{http://elibrary.ru/item.asp?id=14609356}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-42149144759}


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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. Proc. Steklov Inst. Math., 261 (2008), 11–21  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    2. Antontsev S., Shmarev S., “Anisotropic parabolic equations with variable nonlinearity”, Publ. Mat., 53:2 (2009), 355–399  crossref  mathscinet  zmath  isi  elib
    3. Pinasco J.P., “Blow-up for parabolic and hyperbolic problems with variable exponents”, Nonlinear Anal., 71:3-4 (2009), 1094–1099  crossref  mathscinet  zmath  isi
    4. Arnaziane B., Pankratov L., Prytula V., “Homogenization of $p_\varepsilon(x)$-Laplacian in perforated domains with a nonlocal transmission condition”, Comptes Rendus Mécanique, 337:3 (2009), 173–178  crossref  adsnasa  isi
    5. Mashiyev R.A., Buhrii O.M., “Existence of solutions of the parabolic variational inequality with variable exponent of nonlinearity”, J. Math. Anal. Appl., 377:2 (2011), 450–463  crossref  mathscinet  zmath  isi
    6. Antontsev S., Shmarev S., “Parabolic equations with double variable nonlinearities”, Math. Comput. Simulation, 81:10 (2011), 2018–2032  crossref  mathscinet  zmath  isi  elib
    7. Mashiyev R.A., “Three solutions to a Neumann problem for elliptic equations with variable exponent”, Arab. J. Sci. Eng., 36:8 (2011), 1559–1567  crossref  mathscinet  zmath  isi
    8. V. Zh. Sakbaev, “Cauchy problem for degenerating linear differential equations and averaging of approximating regularizations”, Journal of Mathematical Sciences, 213:3 (2016), 287–459  mathnet  crossref  mathscinet
    9. Antontsev S., Shmarev S., “Doubly Degenerate Parabolic Equations with Variable Nonlinearity I: Existence of Bounded Strong Solutions”, Adv. Differ. Equat., 17:11-12 (2012), 1181–1212  mathscinet  zmath  isi
    10. Antontsev S., Chipot M., Shmarev S., “Uniqueness and Comparison Theorems for Solutions of Doubly Nonlinear Parabolic Equations with Nonstandard Growth Conditions”, Commun. Pure Appl. Anal, 12:4 (2013), 1527–1546  crossref  mathscinet  zmath  isi  elib
    11. Simsen J., “A Global Attractor for a P(X)-Laplacian Inclusion”, C. R. Math., 351:3-4 (2013), 87–90  crossref  mathscinet  zmath  isi
    12. E. R. Andriyanova, “Estimates of decay rate for solution to parabolic equation with non-power nonlinearities”, Ufa Math. J., 6:2 (2014), 3–24  mathnet  crossref  elib
    13. E. R. Andriyanova, F. Kh. Mukminov, “Existence of solution for parabolic equation with non-power nonlinearities”, Ufa Math. J., 6:4 (2014), 31–47  mathnet  crossref
    14. Shmarev S., Vdovin V., Vlasov A., “Interfaces in Diffusion-Absorption Processes in Nonhomogeneous Media”, Math. Comput. Simul., 118 (2015), 360–378  crossref  mathscinet  isi  elib
    15. Baravdish G., Svensson O., Astrom F., “on Backward P(X)-Parabolic Equations For Image Enhancement”, Numer. Funct. Anal. Optim., 36:2 (2015), 147–168  crossref  mathscinet  zmath  isi
    16. Abylkairov U.U., Aitzhanov S.E., “Reconstruction of Source Function For Parabolic Equations With Variable Exponents”, Advancements in Mathematical Sciences (Ams 2015), AIP Conference Proceedings, 1676, eds. Ashyralyev A., Malkowsky E., Lukashov A., Basar F., Amer Inst Physics, 2015, 020040  crossref  isi
    17. 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”, Sb. Math., 207:1 (2016), 1–40  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    18. F. Kh. Mukminov, “Uniqueness of the renormalized solutions to the Cauchy problem for an anisotropic parabolic equation”, Ufa Math. J., 8:2 (2016), 44–57  mathnet  crossref  isi  elib
    19. 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  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    20. Almeida R.M.P., Antontsev S.N., Duque J.C.M., “Discrete Solutions For the Porous Medium Equation With Absorption and Variable Exponents”, Math. Comput. Simul., 137:SI (2017), 109–129  crossref  mathscinet  isi  scopus
  • Фундаментальная и прикладная математика
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