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 Izv. RAN. Ser. Mat., 1994, Volume 58, Issue 2, Pages 3–18 (Mi izv800)

Attractor of the generalized semigroup generated by an elliptic equation in a cylindrical domain

A. V. Babin

Abstract: In a domain $\omega\times\mathbf R\subset\mathbf R^{n+1}$ the elliptic system
$$\partial^2_tu+\gamma\partial_tu+a\Delta u-a_0u-f(u)=g \tag{1}$$
is considered with a Neumann boundary condition. $U_+(u_0)$ denotes the set of solutions $u(x,t)$ of this system defined for $t\geqslant 0$, equal to $u_0$ for $t=0$, and bounded in $L_2(\omega)$ uniformly for $t\geqslant 0$.
In the space $H^{3/2}$ of initial data $u_0$ there arises the semigroup $\{S_t\}$, $S_tu_0=\{\upsilon\colon\upsilon=u(t), u\in U_+(u_0)\}$, wherein to the point $u_0$ there is assigned the set $S_tu_0$, i.e., $S_t$ is a multivalued mapping. In the paper it is proved that $\{S_t\}$ has a global attractor $\mathfrak A$. A theorem is proved that
$$\mathfrak A=\{\upsilon\colon\upsilon=u(t), u\in V, t\in\mathbf R\},$$
where $V$ is the set of solutions of the elliptic system, defined and bounded for $t\in\mathbf R$.

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English version:
Russian Academy of Sciences. Izvestiya Mathematics, 1995, 44:2, 207–223

Bibliographic databases:

UDC: 517.95
MSC: Primary 35J55; Secondary 34C35, 47D06

Citation: A. V. Babin, “Attractor of the generalized semigroup generated by an elliptic equation in a cylindrical domain”, Izv. RAN. Ser. Mat., 58:2 (1994), 3–18; Russian Acad. Sci. Izv. Math., 44:2 (1995), 207–223

Citation in format AMSBIB
\Bibitem{Bab94} \by A.~V.~Babin \paper Attractor of the generalized semigroup generated by an elliptic equation in a cylindrical domain \jour Izv. RAN. Ser. Mat. \yr 1994 \vol 58 \issue 2 \pages 3--18 \mathnet{http://mi.mathnet.ru/izv800} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1275899} \zmath{https://zbmath.org/?q=an:0839.35036} \adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?1995IzMat..44..207B} \transl \jour Russian Acad. Sci. Izv. Math. \yr 1995 \vol 44 \issue 2 \pages 207--223 \crossref{https://doi.org/10.1070/IM1995v044n02ABEH001594} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1995RB41200001} 

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