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Mat. Sb., 2014, Volume 205, Number 3, Pages 3–14 (Mi msb8178)  

This article is cited in 18 scientific papers (total in 18 papers)

Existence and uniqueness theorems for solutions of parabolic equations with a variable nonlinearity exponent

Yu. A. Alkhutov, V. V. Zhikov

Vladimir State University

Abstract: The paper is concerned with the solvability of the initial-boundary value problem for second-order parabolic equations with variable nonlinearity exponents. In the model case, this equation contains the $p$-Laplacian with a variable exponent $p(x,t)$. The problem is shown to be uniquely solvable, provided the exponent $p$ is bounded away from both $1$ and $\infty$ and is log-Hölder continuous, and its solution satisfies the energy equality.
Bibliography: 18 titles.

Keywords: parabolic equation, variable nonlinearity exponent, log-Hölder continuity.

Funding Agency Grant Number
Russian Foundation for Basic Research 12-01-00058-а
Ministry of Education and Science of the Russian Federation НШ-3685.2014.1

Author to whom correspondence should be addressed


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English version:
Sbornik: Mathematics, 2014, 205:3, 307–318

Bibliographic databases:

UDC: 517.956.4
MSC: Primary 35K92; Secondary 46E35
Received: 20.09.2012 and 15.01.2014

Citation: Yu. A. Alkhutov, V. V. Zhikov, “Existence and uniqueness theorems for solutions of parabolic equations with a variable nonlinearity exponent”, Mat. Sb., 205:3 (2014), 3–14; Sb. Math., 205:3 (2014), 307–318

Citation in format AMSBIB
\by Yu.~A.~Alkhutov, V.~V.~Zhikov
\paper Existence and uniqueness theorems for solutions of parabolic equations with a~variable nonlinearity exponent
\jour Mat. Sb.
\yr 2014
\vol 205
\issue 3
\pages 3--14
\jour Sb. Math.
\yr 2014
\vol 205
\issue 3
\pages 307--318

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    2. E. Galakhov, O. Salieva, L. Uvarova, “Nonexistence results for some nonlinear elliptic and parabolic inequalities with functional parameters”, Electron. J. Qual. Theory Differ. Equ., 2015, no. 85, 85, 11 pp.  crossref  mathscinet  zmath  isi  scopus
    3. C. O. Alves, S. Shmarev, J. Simsen, M. S. Simsen, “The Cauchy problem for a class of parabolic equations in weighted variable Sobolev spaces: Existence and asymptotic behavior”, J. Math. Anal. Appl., 443:1 (2016), 265–294  crossref  mathscinet  zmath  isi  scopus
    4. A. S. Tersenov, “The one dimensional parabolic $p(x)$-Laplace equation”, NoDEA Nonlinear Differential Equations Appl., 23:3 (2016), Art. 27, 11 pp.  crossref  mathscinet  zmath  isi  scopus
    5. Z. I. Ali, M. Sango, “A note on weak and strong probabilistic solutions for a stochastic quasilinear parabolic equation of generalized polytropic filtration”, Internat. J. Modern Phys. B, 30:28-29 (2016), 1640002, 11 pp.  crossref  mathscinet  zmath  isi  scopus
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    7. E. Galakhov, O. Salieva, L. Uvarova, “Nonexistence results for some nonlinear inequalities with functional parameters”, International Conference of Numerical Analysis and Applied Mathematics 2015 (ICNAAM 2015) (Rhodes, Greece, 22–28 September 2015), AIP Conference Proceedings, 1738, eds. Simos T., Tsitouras C., Amer. Inst. Phys., 2016, 290016  crossref  mathscinet  isi  scopus
    8. A. Youssfi, E. Azroul, B. Lahmi, “Nonlinear parabolic equations with nonstandard growth”, Appl. Anal., 95:12 (2016), 2766–2778  crossref  mathscinet  zmath  isi  scopus
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    10. 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
    11. V. F. Vildanova, F. Kh. Mukminov, “Suschestvovanie slabogo resheniya integro-differentsialnogo uravneniya agregatsii”, Differentsialnye i funktsionalno-differentsialnye uravneniya, SMFN, 63, no. 4, Rossiiskii universitet druzhby narodov, M., 2017, 557–572  mathnet  crossref
    12. F. Kh. Mukminov, “Existence of a renormalized solution to an anisotropic parabolic problem with variable nonlinearity exponents”, Sb. Math., 209:5 (2018), 714–738  mathnet  crossref  crossref  adsnasa  isi  elib
    13. V. F. Vildanova, F. Kh. Mukminov, “Suschestvovanie slabogo resheniya uravneniya agregatsii s $p(\cdot)$-laplasianom”, Matematicheskaya fizika, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 152, VINITI RAN, M., 2018, 34–45  mathnet  mathscinet
    14. Crispo F., Maremonti P., Ruzicka M., “Global l-R-Estimates and Regularizing Effect For Solutions to the P(T, X)-Laplacian Systems”, Adv. Differ. Equat., 24:7-8 (2019), 407–434  mathscinet  isi
    15. F. Kh. Mukminov, “Existence of a Renormalized Solution to an Anisotropic Parabolic Problem for an Equation with Diffuse Measure”, Proc. Steklov Inst. Math., 306 (2019), 178–195  mathnet  crossref  crossref  isi  elib
    16. N. A. Vorobev, F. Kh. Mukminov, “Suschestvovanie renormalizovannogo resheniya parabolicheskoi zadachi v anizotropnykh prostranstvakh Soboleva—Orlicha”, Differentsialnye uravneniya, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 163, VINITI RAN, M., 2019, 39–64  mathnet  mathscinet
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    18. Zakaria A., “Stochastic System For Generalized Polytropic Filtration”, Math. Meth. Appl. Sci., 43:1 (2020), 134–173  crossref  mathscinet  isi
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