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Uspekhi Mat. Nauk, 2009, Volume 64, Issue 3(387), Pages 3–72 (Mi umn9299)  

This article is cited in 2 scientific papers (total in 2 papers)

Large-time asymptotic behaviour of solutions of non-linear Sobolev-type equations

E. I. Kaikinaa, P. I. Naumkina, I. A. Shishmarevb

a National Autonomous University of Mexico, Institute of Mathematics
b M. V. Lomonosov Moscow State University

Abstract: The large-time asymptotic behaviour of solutions of the Cauchy problem is investigated for a non-linear Sobolev-type equation with dissipation. For small initial data the approach taken is based on a detailed analysis of the Green's function of the linear problem and the use of the contraction mapping method. The case of large initial data is also closely considered. In the supercritical case the asymptotic formulae are quasi-linear. The asymptotic behaviour of solutions of a non-linear Sobolev-type equation with a critical non-linearity of the non-convective kind differs by a logarithmic correction term from the behaviour of solutions of the corresponding linear equation. For a critical convective non-linearity, as well as for a subcritical non-convective non-linearity it is proved that the leading term of the asymptotic expression for large times is a self-similar solution. For Sobolev equations with convective non-linearity the asymptotic behaviour of solutions in the subcritical case is the product of a rarefaction wave and a shock wave.
Bibliography: 84 titles.

DOI: https://doi.org/10.4213/rm9299

Full text: PDF file (1047 kB)
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English version:
Russian Mathematical Surveys, 2009, 64:3, 399–468

Bibliographic databases:

UDC: 517.9+535.5
MSC: 35K70, 35B40
Received: 25.07.2008

Citation: E. I. Kaikina, P. I. Naumkin, I. A. Shishmarev, “Large-time asymptotic behaviour of solutions of non-linear Sobolev-type equations”, Uspekhi Mat. Nauk, 64:3(387) (2009), 3–72; Russian Math. Surveys, 64:3 (2009), 399–468

Citation in format AMSBIB
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\pages 3--72
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    This publication is cited in the following articles:
    1. 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
    2. E. I. Kaikina, P. I. Naumkin, I. A. Shishmarev, “Asymptotic expansion of solutions to the periodic problem for a non-linear Sobolev-type equation”, Izv. Math., 77:2 (2013), 313–324  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
  • Успехи математических наук Russian Mathematical Surveys
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