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Izv. RAN. Ser. Mat., 2005, Volume 69, Issue 4, Pages 89–128 (Mi izv649)  

This article is cited in 6 scientific papers (total in 6 papers)

Blow-up of solutions of a class of strongly non-linear dissipative wave equations of Sobolev type with sources

M. O. Korpusov, A. G. Sveshnikov


Abstract: We consider the abstract Cauchy problem for a first-order ordinary differential equation with non-linear operator coefficients. The results are applied to some strongly non-linear dissipative wave equations of Sobolev type. We obtain sufficient conditions for the problem to be globally soluble, as well as sufficient conditions for the solutions to blow up in finite time. These conditions are close to being necessary. Under certain supplementary assumptions on the non-linear operators, we prove that the problem is soluble in any finite cylinder. Under certain conditions on the norm of the initial functions, we prove that the solution of the problem blows up in finite time. We give examples of equations of Sobolev type satisfying these conditions.

DOI: https://doi.org/10.4213/im649

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English version:
Izvestiya: Mathematics, 2005, 69:4, 733–770

Bibliographic databases:

UDC: 517.957
MSC: 78A40, 78A25, 76A10, 76V05, 47J25, 58E05, 35K15, 45K05, 35B45, 35R45, 35J60
Received: 16.11.2004

Citation: M. O. Korpusov, A. G. Sveshnikov, “Blow-up of solutions of a class of strongly non-linear dissipative wave equations of Sobolev type with sources”, Izv. RAN. Ser. Mat., 69:4 (2005), 89–128; Izv. Math., 69:4 (2005), 733–770

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. M. O. Korpusov, A. G. Sveshnikov, “O neobkhodimom i dostatochnom uslovii razrusheniya resheniya smeshannoi kraevoi zadachi dlya odnogo nelineinogo uravneniya sobolevskogo tipa”, Sib. elektron. matem. izv., 2 (2005), 145–155  mathnet  mathscinet  zmath
    2. E. I. Kaikina, P. I. Naumkin, I. A. Shishmarev, “Large-time asymptotic behaviour of solutions of non-linear Sobolev-type equations”, Russian Math. Surveys, 64:3 (2009), 399–468  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    3. E. I. Kaikina, P. I. Naumkin, I. A. Shishmarev, “Periodic Boundary Value Problem for Nonlinear Sobolev-Type Equations”, Funct. Anal. Appl., 44:3 (2010), 171–181  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    4. Hongwei Zhang, Jun Lu, Qingying Hu, “Exponential growth of solution of a strongly nonlinear generalized Boussinesq equation”, Computers & Mathematics with Applications, 2014  crossref  mathscinet  scopus
    5. Le Xuan Truong, Van N.Y., “Exponential Growth With l-P-Norm of Solutions For Nonlinear Heat Equations With Viscoelastic Term”, Appl. Math. Comput., 273 (2016), 656–663  crossref  mathscinet  isi  scopus
    6. Le Xuan Truong, Nguyen Van Y., “On a class of nonlinear heat equations with viscoelastic term”, Comput. Math. Appl., 72:1 (2016), 216–232  crossref  mathscinet  isi  scopus
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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