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

A. V. Babin

Izvestiya. Mathematics , Tome 44 (1995) no. 2, pp. 207-223

Résumé

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 $t\in\mathbf R$.