Abstract:
The asymptotic (t→+∞t→+∞) behavior of solutions of the Cauchy problem is studied for the semilinear parabolic equation
ut=Δu−uβ,t>0,x∈RN;u(0,x)=u0(x)⩾0,x∈RN,
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/(β−1)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 β⩾1+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 u0∈L1(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.
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
\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
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