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Izv. RAN. Ser. Mat., 2014, Volume 78, Issue 5, Pages 91–142 (Mi izv8118)  

This article is cited in 7 scientific papers (total in 7 papers)

Blow-up of solutions of an abstract Cauchy problem for a formally hyperbolic equation with double non-linearity

M. O. Korpusov, A. A. Panin

M. V. Lomonosov Moscow State University, Faculty of Physics

Abstract: We consider an abstract Cauchy problem for a formally hyperbolic equation with double non-linearity. Under certain conditions on the operators in the equation, we prove its local (in time) solubility and give sufficient conditions for finite-time blow-up of solutions of the corresponding abstract Cauchy problem. The proof uses a modification of a method of Levine. We give examples of Cauchy problems and initial-boundary value problems for concrete non-linear equations of mathematical physics.

Keywords: finite-time blow-up, generalized Klein–Gordon equations, non-linear hyperbolic equations, non-linear mixed boundary-value problems, field theory.

Funding Agency Grant Number
Russian Foundation for Basic Research 11-01-12018-офи-м


DOI: https://doi.org/10.4213/im8118

Full text: PDF file (780 kB)
References: PDF file   HTML file

English version:
Izvestiya: Mathematics, 2014, 78:5, 937–985

Bibliographic databases:

UDC: 517.957
MSC: 35B44, 35L20, 35L70
Received: 17.03.2013

Citation: M. O. Korpusov, A. A. Panin, “Blow-up of solutions of an abstract Cauchy problem for a formally hyperbolic equation with double non-linearity”, Izv. RAN. Ser. Mat., 78:5 (2014), 91–142; Izv. Math., 78:5 (2014), 937–985

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. M. Ciarletta, B. Straughan, V. Tibullo, “Explosive instabilities for a generalized second grade fluid”, J. Math. Anal. Appl., 432:2 (2015), 945–953  crossref  mathscinet  zmath  isi  elib  scopus
    2. M. O. Korpusov, “Critical exponents of instantaneous blow-up or local solubility of non-linear equations of Sobolev type”, Izv. Math., 79:5 (2015), 955–1012  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    3. M. Dimova, N. Kolkovska, N. Kutev, “Revised concavity method and application to Klein-Gordon equation”, Filomat, 30:3 (2016), 831–839  crossref  mathscinet  zmath  isi  elib  scopus
    4. M. O. Korpusov, “Solution blow-up in a nonlinear system of equations with positive energy in field theory”, Comput. Math. Math. Phys., 58:3 (2018), 425–436  mathnet  crossref  crossref  isi  elib
    5. 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  mathscinet  isi  elib
    6. 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  isi  elib
    7. M. O. Korpusov, “Blow-up and global solubility in the classical sense of the Cauchy problem for a formally hyperbolic equation with a non-coercive source”, Izv. Math., 84:5 (2020), 930–959  mathnet  crossref  crossref  mathscinet  isi  elib
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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