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

This article is cited in 17 scientific papers (total in 17 papers)

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
\begin{equation} \partial^2_tu+\gamma\partial_tu+a\Delta u-a_0u-f(u)=g \tag{1} \end{equation}
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
Received: 19.10.1992

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
\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
\jour Russian Acad. Sci. Izv. Math.
\yr 1995
\vol 44
\issue 2
\pages 207--223

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    This publication is cited in the following articles:
    1. M. I. Vishik, S. V. Zelik, “The trajectory attractor of a non-linear elliptic system in a cylindrical domain”, Sb. Math., 187:12 (1996), 1755–1789  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. S. V. Zelik, “Boundedness of the solutions of a nonlinear elliptic system in a cylindrical domain”, Math. Notes, 61:3 (1997), 365–369  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    3. A. B. Shapoval, “The integral manifold of a nonlinear elliptic equation in a cylinder”, Math. Notes, 61:3 (1997), 391–395  mathnet  crossref  crossref  mathscinet  zmath  isi
    4. Vadim G. Bondarevsky, “Energetic systems and global attractors for the 3d Navier–Stokes equations”, Nonlinear Analysis: Theory, Methods & Applications, 30:2 (1997), 799  crossref
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    6. M. I. Vishik, S. V. Zelik, “Regular attractor for a non-linear elliptic system in a cylindrical domain”, Sb. Math., 190:6 (1999), 803–834  mathnet  crossref  crossref  mathscinet  zmath  isi
    7. A. Mielke, S. V. Zelik, “Infinite-dimensional trajectory attractors of elliptic boundary-value problems in cylindrical domains”, Russian Math. Surveys, 57:4 (2002), 753–784  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
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  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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