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Sbornik: Mathematics, 2004, Volume 195, Issue 11, Pages 1575–1605
DOI: https://doi.org/10.1070/SM2004v195n11ABEH000858
(Mi sm858)
 

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
References:
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.
Received: 05.06.2003 and 31.05.2004
Bibliographic databases:
UDC: 517.9+535.5
MSC: 76B15, 35B40, 35G10
Language: English
Original paper language: Russian
Citation: E. I. Kaikina, P. I. Naumkin, I. A. Shishmarev, “Cauchy problem for non-linear systems of equations in the critical case”, 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 Sb. Math.
\yr 2004
\vol 195
\issue 11
\pages 1575--1605
\mathnet{http://mi.mathnet.ru//eng/sm858}
\crossref{https://doi.org/10.1070/SM2004v195n11ABEH000858}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2127459}
\zmath{https://zbmath.org/?q=an:1079.35079}
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\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-17744366167}
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  • This publication is cited in the following 1 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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