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Izv. RAN. Ser. Mat., 2020, Volume 84, Issue 3, Pages 15–70 (Mi izv8820)  

This article is cited in 1 scientific paper (total in 1 paper)

Blow-up instability in non-linear wave models with distributed parameters

M. O. Korpusovab, E. A. Ovsyannikovab

a Faculty of Physics, Lomonosov Moscow State University
b Peoples' Friendship University of Russia, Moscow

Abstract: We consider two model non-linear equations describing electric oscillations in systems with distributed parameters on the basis of diodes with non-linear characteristics. We obtain equivalent integral equations for classical solutions of the Cauchy problem and the first and second initial-boundary value problems for the original equations in the half-space $x>0$. Using the contraction mapping principle, we prove the local-in-time solubility of these problems. For one of these equations, we use the Pokhozhaev method of non-linear capacity to deduce a priori bounds giving rise to finite-time blow-up results and obtain upper bounds for the blow-up time. For the other, we use a modification of Levine's method to obtain sufficient conditions for blow-up in the case of sufficiently large initial data and give a lower bound for the order of growth of a functional with the meaning of energy. We also obtain an upper bound for the blow-up time.

Keywords: non-linear equations of Sobolev type, destruction, blow-up, local solubility, non-linear capacity, bounds for the blow-up time.

Funding Agency Grant Number
Ministry of Education and Science of the Russian Federation 5-100
This paper was written with the support of the programme ‘5-100’ of the Peoples' Friendship University of Russia.


DOI: https://doi.org/10.4213/im8820

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English version:
Izvestiya: Mathematics, 2020, 84:3, 449–501

Bibliographic databases:

UDC: 517.538
MSC: 35B44
Received: 05.06.2018
Revised: 20.03.2019

Citation: M. O. Korpusov, E. A. Ovsyannikov, “Blow-up instability in non-linear wave models with distributed parameters”, Izv. RAN. Ser. Mat., 84:3 (2020), 15–70; Izv. Math., 84:3 (2020), 449–501

Citation in format AMSBIB
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\paper Blow-up instability in non-linear wave models with distributed parameters
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\vol 84
\issue 3
\pages 15--70
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\jour Izv. Math.
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\vol 84
\issue 3
\pages 449--501
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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. Jleli M., “Instantaneous Blow-Up of Solutions to the Cauchy Problem For the Fractional Khokhlov-Zabolotskaya Equation”, Open Math., 18 (2020), 1266–1271  crossref  mathscinet  isi
  • Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya Izvestiya: Mathematics
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