Regular attractor for a~non-linear elliptic system in a~cylindrical domain

M. I. Vishik ; S. V. Zelik

Sbornik. Mathematics, Tome 190 (1999) no. 6, pp. 803-834

Résumé

The system of second-order elliptic equations \begin{equation} a(\partial_t^2u+\Delta_xu)-\gamma\partial_tu-f(u)=g(t), \quad u\big|_{\partial\omega}=0, \enskip u\big|_{t=0}=u_0, \enskip (t,x)\in\Omega _+, \tag{1} \end{equation} is considered in the half-cylinder $\Omega_+=\mathbb R_+\times\omega$, $\omega\subset\mathbb R^n$. Here $u=(u^1,\dots,u^k)$ is an unknown vector-valued function, $a$ and $\gamma$ are fixed positive-definite self-adjoint $(k\times k)$-matrices, $f$ and $g(t)=g(t,x)$ are fixed functions. It is proved under certain natural conditions on the matrices $a$ and $\gamma$, the non-linear function $f$, and the right-hand side $g$ that the boundary-value problem (1) has a unique solution in the space $W^{2,p}_{\mathrm{loc}}(\Omega_+,\mathbb R^k)$, $p>(n+1)/2$, that is bounded as $t\to\infty$. Moreover, it is established that the problem (1) is equivalent in the class of such solutions to an evolution problem in the space of “initial data” $u_0\in V_0\equiv\operatorname{Tr}_{t=0}W^{2,p}_{\mathrm{loc}}(\Omega_+,\mathbb R^k)$. In the potential case $(f=\nabla _x P$,  $g(t,x)\equiv g(x))$ it is shown that the semigroup $S_t\colon V_0\to V_0$ generated by (1) possesses an attractor in the space $V_0$ which is generically the union of finite-dimensional unstable manifolds $\mathscr M^+(z_i)$ corresponding to the equilibria $z_i$ of $S_t$ $(S_tz_i=z_i)$. In addition, an explicit formula for the dimensions of these manifolds is obtained.