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Mat. Sb., 2004, Volume 195, Number 11, Pages 31–62 (Mi msb858)  

This article is cited in 1 scientific paper (total in 1 paper)

Cauchy problem for non-linear systems of equations in the critical case

E. I. Kaikina, P. I. Naumkin, I. A. Shishmarev

M. V. Lomonosov Moscow State University

Abstract: The large-time asymptotic behaviour is studied for a system of non-linear evolution dissipative equations
\begin{gather*} u_t+\mathscr N(u,u)+\mathscr Lu=0, \qquad x\in\mathbb R^n, \quad t>0,
u(0,x)=\widetilde u(x), \qquad x\in\mathbb R^n, \end{gather*}
where $\mathscr L$ is a linear pseudodifferential operator $\mathscr Lu=\overline{\mathscr F}_{\xi\to x}(L(\xi)\widehat u(\xi))$ and the non-linearity $\mathscr N$ is a quadratic pseudodifferential operator
$$ \mathscr N(u,u)=\overline{\mathscr F}_{\xi\to x}\sum_{k,l=1}^m\int_{\mathbb R^n}A^{kl}(t,\xi,y)\widehat u_k(t,\xi-y)\widehat u_l(t,y) dy, $$
where $\widehat u\equiv\mathscr F_{x\to\xi}u$ is the Fourier transform. Under the assumptions that the initial data $\widetilde u\in\mathbf H^{\beta,0}\cap\mathbf H^{0,\beta}$, $\beta>n/2$ are sufficiently small, where
$$ \mathbf H^{n,m}=\{\phi\in\mathbf L^2:\|\langle x\rangle^m\langle i\partial_x\rangle^n\phi(x)\|_{\mathbf L^2}<\infty\}, \qquad \langle x\rangle=\sqrt{1+x^2} , $$
is a Sobolev weighted space, and that the total mass vector $\displaystyle M=\int\widetilde u(x) dx\ne0$ is non-zero it is proved that the leading term in the large-time asymptotic expansion of solutions in the critical case is a self-similar solution defined uniquely by the total mass vector $M$ of the initial data.

DOI: https://doi.org/10.4213/sm858

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English version:
Sbornik: Mathematics, 2004, 195:11, 1575–1605

Bibliographic databases:

UDC: 517.9+535.5
MSC: 76B15, 35B40, 35G10
Received: 05.06.2003 and 31.05.2004

Citation: E. I. Kaikina, P. I. Naumkin, I. A. Shishmarev, “Cauchy problem for non-linear systems of equations in the critical case”, Mat. Sb., 195:11 (2004), 31–62; Sb. Math., 195:11 (2004), 1575–1605

Citation in format AMSBIB
\Bibitem{KaiNauShi04}
\by E.~I.~Kaikina, P.~I.~Naumkin, I.~A.~Shishmarev
\paper Cauchy problem for non-linear systems of equations in the critical case
\jour Mat. Sb.
\yr 2004
\vol 195
\issue 11
\pages 31--62
\mathnet{http://mi.mathnet.ru/msb858}
\crossref{https://doi.org/10.4213/sm858}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2127459}
\zmath{https://zbmath.org/?q=an:1079.35079}
\transl
\jour Sb. Math.
\yr 2004
\vol 195
\issue 11
\pages 1575--1605
\crossref{https://doi.org/10.1070/SM2004v195n11ABEH000858}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000228585900003}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-17744366167}


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    This publication is cited in the following articles:
    1. Hayashi N., Kaikina E.I., Naumkin P.I., Shishmarev I.A., Asymptotics for dissipative nonlinear equations, Lecture Notes in Math., 1884, Springer-Verlag, Berlin–Heidelberg, 2006, xii+557 pp.  crossref  mathscinet  zmath  isi
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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