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Zh. Vychisl. Mat. Mat. Fiz., 2007, Volume 47, Number 2, Pages 245–255 (Mi zvmmf332)  

This article is cited in 16 scientific papers (total in 16 papers)

Cauchy problem for a quasilinear parabolic equation with a source term and an inhomogeneous density

A. V. Martynenko, A. F. Tedeev

Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, ul. R. Luxemburg 74, Donetsk, 83114, Ukraine

Abstract: The following quasilinear parabolic equation with a source term and an inhomogeneous density is considered:
$$ \rho(x)\frac{\partial u}{\partial t}=\operatorname{div}(u^{m-1}|Du|^{\lambda-1}Du)+u^p. $$
The conditions on the parameters of the problem are found under which the solution to the Cauchy problem blows up in a finite time. A sharp universal (i.e., independent of the initial function) estimate of the solution near the blowup time is obtained.

Key words: inhomogeneous density, degenerate parabolic equation, blowup regime.

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English version:
Computational Mathematics and Mathematical Physics, 2007, 47:2, 238–248

Bibliographic databases:

UDC: 519.633
Received: 26.05.2006

Citation: A. V. Martynenko, A. F. Tedeev, “Cauchy problem for a quasilinear parabolic equation with a source term and an inhomogeneous density”, Zh. Vychisl. Mat. Mat. Fiz., 47:2 (2007), 245–255; Comput. Math. Math. Phys., 47:2 (2007), 238–248

Citation in format AMSBIB
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\by A.~V.~Martynenko, A.~F.~Tedeev
\paper Cauchy problem for a~quasilinear parabolic equation with a~source term and an inhomogeneous density
\jour Zh. Vychisl. Mat. Mat. Fiz.
\yr 2007
\vol 47
\issue 2
\pages 245--255
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\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2351815}
\zmath{https://zbmath.org/?q=an:05200978}
\transl
\jour Comput. Math. Math. Phys.
\yr 2007
\vol 47
\issue 2
\pages 238--248
\crossref{https://doi.org/10.1134/S096554250702008X}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-33947126955}


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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. A. V. Martynenko, A. F. Tedeev, “On the behavior of solutions to the Cauchy problem for a degenerate parabolic equation with inhomogeneous density and a source”, Comput. Math. Math. Phys., 48:7 (2008), 1145–1160  mathnet  crossref  isi  elib
    2. Zheng Sining, Wang Chunpeng, “Large time behaviour of solutions to a class of quasilinear parabolic equations with convection terms”, Nonlinearity, 21:9 (2008), 2179–2200  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    3. Cao Yang, Yin Jingxue, Wang Chunpeng, “Cauchy problems of semilinear pseudo-parabolic equations”, J. Differential Equations, 246:12 (2009), 4568–4590  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    4. Cianci P., Martynenko A.V., Tedeev A.F., “The blow-up phenomenon for degenerate parabolic equations with variable coefficients and nonlinear source”, Nonlinear Anal., 73:7 (2010), 2310–2323  crossref  mathscinet  zmath  isi  elib  scopus
    5. Novruzov E., “On existence and nonexistence of the positive solutions of non-Newtonian filtration equation”, Commun. Pure Appl. Anal., 10:2 (2011), 719–730  crossref  mathscinet  zmath  isi  elib  scopus
    6. Iagar R.G., Reyes G., Sanchez A., “Radial Equivalence of Nonhomogeneous Nonlinear Diffusion Equations”, Acta Appl. Math., 123:1 (2013), 53–72  crossref  mathscinet  zmath  isi  elib  scopus
    7. de Pablo A., Reyes G., Sanchez A., “The Cauchy Problem for a Nonhomogeneous Heat Equation with Reaction”, Discret. Contin. Dyn. Syst., 33:2 (2013), 643–662  crossref  mathscinet  zmath  isi  scopus
    8. Martynenko A.V., “Global Solvability for Quasilinear Parabolic Equation with Inhomogeneous Density and a Source”, Appl. Anal., 92:9 (2013), 1863–1888  crossref  mathscinet  zmath  isi  elib  scopus
    9. Martynenko A.V. Tedeev A.F. Shramenko V.N., “On the Behavior of Solutions of the Cauchy Problem for a Degenerate Parabolic Equation with Source in the Case Where the Initial Function Slowly Vanishes”, Ukr. Math. J., 64:11 (2013), 1698–1715  crossref  mathscinet  zmath  isi  elib  scopus
    10. Li Zh., Mu Ch., Du W., “Critical Fujita Exponent for a Fast Diffusive Equation with Variable Coefficients”, Bull. Korean. Math. Soc., 50:1 (2013), 105–116  crossref  mathscinet  zmath  isi  elib  scopus
    11. Zheng P., Mu Ch., “Global Existence, Large Time Behavior, and Life Span For a Degenerate Parabolic Equation With Inhomogeneous Density and Source”, Z. Angew. Math. Phys., 65:3 (2014), 471–486  crossref  mathscinet  zmath  isi  elib  scopus
    12. Li Zh., Du W., “Life Span and Secondary Critical Exponent For Degenerate and Singular Parabolic Equations”, Ann. Mat. Pura Appl., 193:2 (2014), 501–515  crossref  mathscinet  zmath  isi  elib  scopus
    13. Li N. Wang L. Mu Ch. Zheng P., “Disappearance and Global Existence of Interfaces For a Doubly Degenerate Parabolic Equation With Variable Coefficient”, Math. Meth. Appl. Sci., 38:8 (2015), 1465–1471  crossref  mathscinet  zmath  isi  elib  scopus
    14. Aripov M.M., Matyakubov A.S., “Self-similar solutions of a cross-diffusion parabolic system with variable density: explicit estimates and asymptotic behaviour”, Nanosyst.-Phys. Chem. Math., 8:1 (2017), 5–12  crossref  mathscinet  isi
    15. Leng Ya., Nie Yu., Zhou Q., “Asymptotic Behavior of Solutions to a Class of Coupled Nonlinear Parabolic Systems”, Bound. Value Probl., 2019, 68  crossref  mathscinet  isi  scopus
    16. Z. V. Besaeva, A. F. Tedeev, “Skorost ubyvaniya massy resheniya zadachi Koshi dvazhdy nelineinogo parabolicheskogo uravneniya s absorbtsiei”, Vladikavk. matem. zhurn., 22:1 (2020), 12–32  mathnet  crossref
  • Журнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
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