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Izv. RAN. Ser. Mat., 2018, Volume 82, Issue 2, Pages 43–78 (Mi izv8579)  

This article is cited in 5 scientific papers (total in 5 papers)

Blow-up of solutions of a full non-linear equation of ion-sound waves in a plasma with non-coercive non-linearities

M. O. Korpusova, D. V. Lukyanenkoa, A. A. Paninab, E. V. Yushkovac

a Lomonosov Moscow State University, Faculty of Physics
b Nikol'skii Mathematical Institute of Peoples’ Friendship University of Russia
c Space Research Institute, Russian Academy of Sciences, Moscow

Abstract: We consider a series of initial-boundary value problems for the equation of ion-sound waves in a plasma. For each of them we prove the local (in time) solubility and perform an analytical-numerical study of the blow-up of solutions. We use the method of test functions to obtain sufficient conditions for finite-time blow-up and an upper bound for the blow-up time. In concrete numerical examples we improve these bounds numerically using the mesh refinement method. Thus the analytical and numerical parts of the investigation complement each other. The time interval for the numerical modelling is chosen in accordance with the analytically obtained upper bound for the blow-up time. In return, numerical calculations specify the moment and pattern of this blow-up.

Keywords: blow-up of a solution, non-linear initial-boundary value problem, Sobolev-type equations, exponential non-linearity, Richardson extrapolation.

Funding Agency Grant Number
Russian Foundation for Basic Research 15-01-03524
16-32-00011
16-01-00755
16-01-00437
14-01-00182
14-01-00208
This paper was written with the support of the Russian Foundation for Basic Research (grants nos. 15-01-03524, 16-32-00011, 16-01-00755, 16-01-00437, 14-01-00182 and 14-01-00208).


DOI: https://doi.org/10.4213/im8579

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English version:
Izvestiya: Mathematics, 2018, 82:2, 283–317

Bibliographic databases:

UDC: 517.957+519.6
MSC: 35B44, 35L35, 35Q60, 76X05
Received: 11.06.2016

Citation: M. O. Korpusov, D. V. Lukyanenko, A. A. Panin, E. V. Yushkov, “Blow-up of solutions of a full non-linear equation of ion-sound waves in a plasma with non-coercive non-linearities”, Izv. RAN. Ser. Mat., 82:2 (2018), 43–78; Izv. Math., 82:2 (2018), 283–317

Citation in format AMSBIB
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\paper Blow-up of solutions of a~full non-linear equation of ion-sound waves
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\jour Izv. RAN. Ser. Mat.
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\vol 82
\issue 2
\pages 43--78
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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. M. O. Korpusov, D. V. Lukyanenko, A. A. Panin, G. I. Shlyapugin, “On the blow-up phenomena for a 1-dimensional equation of ion sound waves in a plasma: analytical and numerical investigation”, Math. Methods Appl. Sci., 41:8 (2018), 2906–2929  crossref  mathscinet  zmath  isi
    2. I. I. Kolotov, A. A. Panin, “On Nonextendable Solutions and Blow-Ups of Solutions of Pseudoparabolic Equations with Coercive and Constant-Sign Nonlinearities: Analytical and Numerical Study”, Math. Notes, 105:5 (2019), 694–706  mathnet  crossref  crossref  isi  elib
    3. M. O. Korpusov, A. K. Matveeva, D. V. Lukyanenko, “Diagnostika mgnovennogo razrusheniya resheniya v nelineinom uravnenii teorii voln v poluprovodnikakh”, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 12:4 (2019), 104–113  mathnet  crossref
    4. A. A. Panin, G. I. Shlyapugin, “Local Solvability and Global Unsolvability of a Model of Ion-Sound Waves in a Plasma”, Math. Notes, 107:3 (2020), 464–477  mathnet  crossref  crossref  isi
    5. M. O. Korpusov, E. A. Ovsyannikov, “Blow-up instability in non-linear wave models with distributed parameters”, Izv. Math., 84:3 (2020), 449–501  mathnet  crossref  crossref
  • Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya Izvestiya: Mathematics
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