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Mathematics of the USSR-Sbornik, 1986, Volume 54, Issue 2, Pages 421–455
DOI: https://doi.org/10.1070/SM1986v054n02ABEH002979
(Mi sm1946)
 

This article is cited in 81 scientific papers (total in 81 papers)

On asymptotic “eigenfunctions” of the Cauchy problem for a nonlinear parabolic equation

V. A. Galaktionov, S. P. Kurdyumov, A. A. Samarskii
References:
Abstract: The asymptotic (t+t+) behavior of solutions of the Cauchy problem is studied for the semilinear parabolic equation
ut=Δuuβ,t>0, xRN;u(0,x)=u0(x)0,xRN,
where β=const>1 and u0(x)0 as |x|+. The existence is established of an infinite collection (a continuum) of distinct self-similar solutions of the form uA(t,x)=(T+t)1/(β1)θA(ξ), ξ=|x|/(T+t)1/2, where the function θA>0 satisfies an ordinary differential equation. Conditions for the asymptotic stability of these solutions are established. It is shown that for β1+2/N there exist solutions of the problem whose behavior as t+ is described by approximate self-similar solutions (ap.s.-s.s.'s) ua(t,x) which in the case β>1+2/N coincide with a family of self-similar solutions of the heat equation (ua)t=Δua, while for β=1+2/N and u0L1(RN) the ap.s.-s.s. has the form ua=[(T+t)ln(T+t)]N/2cNexp(|x|2/4(T+t)), where cN=(N/2)N/2(1+2/N)N2/4.
Figures: 2.
Bibliography: 78 titles.
Received: 23.07.1984
Bibliographic databases:
UDC: 517.956
MSC: Primary 35K55, 35K15; Secondary 35K05, 35B35
Language: English
Original paper language: Russian
Citation: V. A. Galaktionov, S. P. Kurdyumov, A. A. Samarskii, “On asymptotic “eigenfunctions” of the Cauchy problem for a nonlinear parabolic equation”, Math. USSR-Sb., 54:2 (1986), 421–455
Citation in format AMSBIB
\Bibitem{GalKurSam85}
\by V.~A.~Galaktionov, S.~P.~Kurdyumov, A.~A.~Samarskii
\paper On asymptotic ``eigenfunctions'' of the Cauchy problem for a~nonlinear parabolic equation
\jour Math. USSR-Sb.
\yr 1986
\vol 54
\issue 2
\pages 421--455
\mathnet{http://mi.mathnet.ru/eng/sm1946}
\crossref{https://doi.org/10.1070/SM1986v054n02ABEH002979}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=788082}
\zmath{https://zbmath.org/?q=an:0607.35049}
Linking options:
  • https://www.mathnet.ru/eng/sm1946
  • https://doi.org/10.1070/SM1986v054n02ABEH002979
  • https://www.mathnet.ru/eng/sm/v168/i4/p435
    Erratum
    • Letter to the Editor
      V. A. Galaktionov, S. P. Kurdyumov, A. A. Samarskii
      Mat. Sb. (N.S.), 1986, 131(173):3(11), 413
    This publication is cited in the following 81 articles:
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    2. V. L. Natyaganov, Yu. D. Skobennikova, “A solution to heat equation with exacerbation and stopped heat wave”, Moscow University Mechanics Bulletin, 77:5 (2022), 151–153  mathnet  crossref  elib
    3. Sarah Otsmane, “Asymptotically self-similar global solutions for a complex-valued quadratic heat equation with a generalized kernel”, Bol. Soc. Mat. Mex., 27:2 (2021)  crossref
    4. Mirsaid M. Aripov, Shakhlo A. Sadullayeva, Maftuha Z. Sayfullayeva, INTERNATIONAL CONFERENCE ON ANALYSIS AND APPLIED MATHEMATICS (ICAAM 2020), 2325, INTERNATIONAL CONFERENCE ON ANALYSIS AND APPLIED MATHEMATICS (ICAAM 2020), 2021, 020064  crossref
    5. Amel Chouichi, Sarah Otsmane, Slim Tayachi, “Large time behavior of solutions for a complex-valued quadratic heat equation”, Nonlinear Differ. Equ. Appl, 2015  crossref  mathscinet
    6. Jakhongir R. Raimbekov, “The properties of the solutions for Cauchy problem of nonlinear parabolic equations in non-divergent form with density”, Zhurn. SFU. Ser. Matem. i fiz., 8:2 (2015), 192–200  mathnet
    7. Lucas C. F. Ferreira, Marcelo F. Furtado, Everaldo S. Medeiros, “Existence and multiplicity of self-similar solutions for heat equations with nonlinear boundary conditions”, Calc. Var., 54:4 (2015), 4065  crossref
    8. Proc. Steklov Inst. Math., 283 (2013), 44–74  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    9. Gordon P.V., Muratov C.B., “Self-Similarity and Long-Time Behavior of Solutions of the Diffusion Equation with Nonlinear Absorption and a Boundary Source”, Netw. Heterog. Media, 7:4, SI (2012), 767–780  crossref  mathscinet  zmath  isi
    10. Liskevich V., Shishkov A., Sobol Z., “Singular Solutions to the Heat Equations with Nonlinear Absorption and Hardy Potentials”, Commun. Contemp. Math., 14:2 (2012), 1250013  crossref  mathscinet  zmath  isi
    11. Kenji Nishihara, “Decay Properties for the Damped Wave Equation with Space Dependent Potential and Absorbed Semilinear Term”, Comm. in Partial Differential Equations, 35:8 (2010), 1402  crossref  mathscinet  zmath
    12. M. Hamza, “Asymptotically self-similar solutions of the damped wave equation”, Nonlinear Analysis: Theory, Methods & Applications, 73:9 (2010), 2897  crossref  mathscinet  zmath
    13. Bernoff A.J., Witelski T.P., “Stability and Dynamics of Self-Similarity in Evolution Equations”, J. Eng. Math., 66:1-3 (2010), 11–31  crossref  mathscinet  zmath  isi  elib
    14. 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
    15. Sachdev* P.L., Srinivasa Rao Ch., Springer Monographs in Mathematics, Large Time Asymptotics for Solutions of Nonlinear Partial Differential Equations, 2009, 129  crossref
    16. Elena I. Kaikina, “Global behavior for Whitham type equations on a finite interval”, Calc Var, 33:1 (2008), 113  crossref  mathscinet  zmath  isi
    17. Andrey Shishkov, Laurent Véron, “Singular solutions of some nonlinear parabolic equations with spatially inhomogeneous absorption”, Calc Var, 33:3 (2008), 343  crossref  mathscinet  zmath  isi
    18. ELENA I. KAIKINA, “FRACTIONAL Landau–Ginzburg EQUATIONS ON A SEGMENT”, Commun. Contemp. Math, 10:06 (2008), 1151  crossref  mathscinet  zmath
    19. Kaikina E.I., Naumkin P.I., Shishmarev I.A., “Asymptotic Behavior of Solutions to a Boundary Value Problem for a Nonlinear Equation with a Fractional Derivative”, Dokl. Math., 78:1 (2008), 485–487  crossref  mathscinet  zmath  isi  elib
    20. N. Hayashi, E. I. Kaikina, P. I. Naumkin, “Asymptotics for nonlinear damped wave equations with large initial data”, Sib. elektron. matem. izv., 4 (2007), 249–277  mathnet  mathscinet  zmath
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