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Matematicheskie Zametki, 2022, Volume 111, Issue 3, Pages 459–475
DOI: https://doi.org/10.4213/mzm13256
(Mi mzm13256)
 

This article is cited in 3 scientific papers (total in 3 papers)

Solvability and Blow-Up of Weak Solutions of Cauchy Problems for $(3+1)$-Dimensional Equations of Drift Waves in a Plasma

R. S. Shafir

Lomonosov Moscow State University
Full-text PDF (584 kB) Citations (3)
References:
Abstract: In this paper, two Cauchy problems that contain different nonlinearities $|u|^q$ and $(\partial/\partial t)|u|^q$ are studied. The differential operator in these problems is the same. It is defined by the formula $\mathfrak{M}_{x,t}:=(\partial^2/\partial t^2)\Delta_{\perp}+ \partial^2/\partial x_3^2$. The problems have a concrete physical meaning, namely, they describe drift waves in a magnetically active plasma. Conditions are found under which weak generalized solutions of these Cauchy problems exist and also under which weak solutions of the same Cauchy problems blow up. However, the question of the uniqueness of weak generalized solutions of Cauchy problems remains open, because uniqueness conditions have not been found.
Keywords: Sobolev-type nonlinear equations, blow-up, local solvability, nonlinear capacity.
Received: 10.08.2021
Revised: 20.10.2021
English version:
Mathematical Notes, 2022, Volume 111, Issue 3, Pages 484–497
DOI: https://doi.org/10.1134/S0001434622030166
Bibliographic databases:
Document Type: Article
UDC: 517.538
Language: Russian
Citation: R. S. Shafir, “Solvability and Blow-Up of Weak Solutions of Cauchy Problems for $(3+1)$-Dimensional Equations of Drift Waves in a Plasma”, Mat. Zametki, 111:3 (2022), 459–475; Math. Notes, 111:3 (2022), 484–497
Citation in format AMSBIB
\Bibitem{Sha22}
\by R.~S.~Shafir
\paper Solvability and Blow-Up of Weak Solutions of Cauchy Problems for $(3+1)$-Dimensional Equations of Drift Waves in a Plasma
\jour Mat. Zametki
\yr 2022
\vol 111
\issue 3
\pages 459--475
\mathnet{http://mi.mathnet.ru/mzm13256}
\crossref{https://doi.org/10.4213/mzm13256}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4461276}
\transl
\jour Math. Notes
\yr 2022
\vol 111
\issue 3
\pages 484--497
\crossref{https://doi.org/10.1134/S0001434622030166}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85129252876}
Linking options:
  • https://www.mathnet.ru/eng/mzm13256
  • https://doi.org/10.4213/mzm13256
  • https://www.mathnet.ru/eng/mzm/v111/i3/p459
  • This publication is cited in the following 3 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Математические заметки Mathematical Notes
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    Full-text PDF :28
    References:60
    First page:10
     
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