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Tr. Mat. Inst. Steklova, 2008, Volume 261, Pages 16–25 (Mi tm736)  

This article is cited in 23 scientific papers (total in 23 papers)

Extinction of Solutions of Parabolic Equations with Variable Anisotropic Nonlinearities

S. N. Antontsevab, S. I. Shmarevc

a Departamento de Matemática, Universidade da Beira Interior
b Departamento de Matemática Aplicada, Universidad Complutence
c Departaménto de Matemáticas, Universidad de Oviedo

Abstract: We study the Dirichlet problem for a class of nonlinear parabolic equations with nonstandard anisotropic growth conditions that generalize the evolutional $p(x,t)$-Laplacian. We study the property of extinction of solutions in finite time. In particular, we show that the extinction may take place even in the borderline case when the equation becomes linear as $t\to\infty$.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2008, 261, 11–21

Bibliographic databases:

UDC: 517.956.4
Received in March 2007

Citation: S. N. Antontsev, S. I. Shmarev, “Extinction of Solutions of Parabolic Equations with Variable Anisotropic Nonlinearities”, Differential equations and dynamical systems, Collected papers, Tr. Mat. Inst. Steklova, 261, MAIK Nauka/Interperiodica, Moscow, 2008, 16–25; Proc. Steklov Inst. Math., 261 (2008), 11–21

Citation in format AMSBIB
\by S.~N.~Antontsev, S.~I.~Shmarev
\paper Extinction of Solutions of Parabolic Equations with Variable Anisotropic Nonlinearities
\inbook Differential equations and dynamical systems
\bookinfo Collected papers
\serial Tr. Mat. Inst. Steklova
\yr 2008
\vol 261
\pages 16--25
\publ MAIK Nauka/Interperiodica
\publaddr Moscow
\jour Proc. Steklov Inst. Math.
\yr 2008
\vol 261
\pages 11--21

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    2. Antontsev S., Shmarev S., “Anisotropic parabolic equations with variable nonlinearity”, Publ. Mat., 53:2 (2009), 355–399  crossref  mathscinet  zmath  isi  elib  scopus
    3. Antontsev S., Shmarev S., “Localization of solutions of anisotropic parabolic equations”, Nonlinear Anal., 71:12 (2009), e725–e737  crossref  mathscinet  zmath  isi  scopus
    4. Antontsev S., Shmarev S., “Vanishing solutions of anisotropic parabolic equations with variable nonlinearity”, J. Math. Anal. Appl., 361:2 (2010), 371–391  crossref  mathscinet  zmath  isi  elib  scopus
    5. Proc. Steklov Inst. Math., 270 (2010), 27–42  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    6. Antontsev S., Shmarev S., “Blow-up of solutions to parabolic equations with nonstandard growth conditions”, J. Comput. Appl. Math., 234:9 (2010), 2633–2645  crossref  mathscinet  zmath  isi  elib  scopus
    7. Antontsev S., “Wave equation with $p(x,t)$-Laplacian and damping term: Blow-up of solutions”, Comptes Rendus Mécanique, 339:12 (2011), 751–755  crossref  mathscinet  adsnasa  isi  scopus
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    9. Vétois J., “Strong maximum principles for anisotropic elliptic and parabolic equations”, Adv. Nonlinear Stud., 12:1 (2012), 101–114  crossref  mathscinet  zmath  isi  elib  scopus
    10. 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
    11. Akagi G., “Doubly Nonlinear Parabolic Equations Involving Variable Exponents”, Discret. Contin. Dyn. Syst.-Ser. S, 7:1 (2014), 1–16  crossref  mathscinet  zmath  isi  elib
    12. 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
    13. Crispo F., “a Note on the Existence and Uniqueness of Time-Periodic Electro-Rheological Flows”, Acta Appl. Math., 132:1, SI (2014), 237–250  crossref  mathscinet  zmath  isi  scopus
    14. Chai X. Li H. Niu W., “Large Time Behavior For P(X)-Laplacian Equations With Irregular Data”, Electron. J. Differ. Equ., 2015, 61  mathscinet  zmath  isi  elib
    15. 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
    16. Liu B., Yang J., “Blow-up properties in the parabolic problems with anisotropic nonstandard growth conditions”, Z. Angew. Math. Phys., 67:1 (2016), UNSP 13  crossref  mathscinet  isi  elib  scopus
    17. Antontsev S., Shmarev S., Simsen J., Simsen M.S., Nonlinear Anal.-Theory Methods Appl., 134 (2016), 31–54  crossref  mathscinet  zmath  isi  scopus
    18. Bokalo M.M., Ilnytska O.V., “Problems For Parabolic Equations With Variable Exponents of Nonlinearity and Time Delay”, Appl. Anal., 96:7 (2017), 1240–1254  crossref  mathscinet  zmath  isi  scopus
    19. Liu B., Xin Q., Dong M., “Blow-Up Analyses in Parabolic Equations With Anisotropic Nonstandard Damping Source”, J. Math. Anal. Appl., 458:1 (2018), 242–264  crossref  mathscinet  zmath  isi  scopus
    20. Giacomoni J., “Quasilinear Parabolic Problem With Variable Exponent: Qualitative Analysis and Stabilization”, Commun. Contemp. Math., 20:8 (2018), 1750065  crossref  mathscinet  zmath  isi  scopus
    21. Liu B., Dong M., Li F., “Singular Solutions in Nonlinear Parabolic Equations With Anisotropic Nonstandard Growth Conditions”, J. Math. Phys., 59:12 (2018), 121504  crossref  mathscinet  zmath  isi  scopus
    22. Liu B., Dong M., “A Nonlinear Diffusion Problem With Convection and Anisotropic Nonstandard Growth Conditions”, Nonlinear Anal.-Real World Appl., 48 (2019), 383–409  crossref  mathscinet  isi  scopus
    23. 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  isi
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