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Izv. RAN. Ser. Mat., 2012, Volume 76, Issue 3, Pages 139–156 (Mi izv6847)  

This article is cited in 4 scientific papers (total in 4 papers)

The Cauchy problem for a degenerate parabolic equation with inhomogeneous density and source in the class of slowly decaying initial data

A. V. Martynenkoa, A. F. Tedeevb, V. N. Shramenkoc

a Lugansk Taras Shevchenko National University
b Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine
c National Technical University of Ukraine "Kiev Polytechnic Institute"

Abstract: Given a degenerate parabolic equation of the form $\rho(x) u_t=\operatorname{div}(u^{m-1}|Du|^{\lambda-1}Du)+\rho(x)u^p$ with a source and inhomogeneous density, we consider the Cauchy problem with an initial function slowly tending to zero as $|x| \to \infty$. We find conditions for the global-in-time existence or non-existence of solutions of this problem. These conditions depend essentially on the behaviour of the initial data as $|x|\to \infty$. In the case of global solubility we obtain a sharp estimate of the solution for large values of time.

Keywords: inhomogeneous density, degenerate parabolic equation, blow-up, slowly decaying initial function.

DOI: https://doi.org/10.4213/im6847

Full text: PDF file (578 kB)
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English version:
Izvestiya: Mathematics, 2012, 76:3, 563–580

Bibliographic databases:

UDC: 517.946
MSC: 35K65, 35K15, 35B44
Received: 31.01.2011

Citation: A. V. Martynenko, A. F. Tedeev, V. N. Shramenko, “The Cauchy problem for a degenerate parabolic equation with inhomogeneous density and source in the class of slowly decaying initial data”, Izv. RAN. Ser. Mat., 76:3 (2012), 139–156; Izv. Math., 76:3 (2012), 563–580

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. A. V. Martynenko, A. F. Tedeev, V. N. Shramenko, “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”, Ukrainian Math. J., 64:11 (2013), 1698–1715  crossref  mathscinet  zmath  isi  scopus
    2. Pan Zheng, Chunlai Mu, “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  scopus
    3. Xie Li, Zhaoyin Xiang, “Existence and nonexistence of local/global solutions for a nonhomogeneous heat equation”, Commun. Pure Appl. Anal., 13:4 (2014), 1465–1480  crossref  mathscinet  zmath  isi  scopus
    4. Pan Zheng, Chunlai Mu, Iftikhar Ahmed, “Cauchy problem for the non-Newtonian polytropic filtration equation with a localized reaction”, Appl. Anal., 94:1 (2015), 153–168  crossref  mathscinet  zmath  isi  scopus
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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