The trajectory attractor of a non-linear elliptic system in a cylindrical domain

In the half-cylinder $\Omega _+=\mathbb R_+\times \omega$, $\omega \in \mathbb R^n$, we study a second-order system of elliptic equations containing a non-linear function $f(u,x_0,x')=(f^1,\dots ,f^k)$ and right-hand side $g(x_0,x')=(g^1,\dots ,g^k)$, $x_0\in \mathbb R_+$, $x'\in \omega$. If these functions satisfy certain conditions, then it is proved that the first boundary-value problem for this system has at least one solution belonging to the space $[H_{2,p}^{\text {loc}}(\Omega _+)]^k$, $p>n+1$. We study the behaviour of the solutions $u(x_0,x')$ of this system a $x_0\to +\infty$. Along with the original system we study the family of systems obtained from it through shifting with respect to $x_0$ by all $\forall \,h$, $h\geqslant 0$. A semigroup $\{T(h),\ h\geqslant 0\}$, $T(h)u(x_0,\,\cdot \,)=u(x_0+h,\,\cdot \,)$ acts on the set of solutions $K^+$ of these systems of equations. It is proved that this semigroup has a trajectory attractor $\mathbb A$ consisting of the solutions $v(x_0,x')$ in $K^+$ that admit a bounded extension to the entire cylinder $\Omega =\mathbb R\times \omega$. Solutions $u(x_0,x')\in K^+$ are attracted by the attractor $\mathbb A$ as $x_0\to +\infty$. We give a number of applications and consider some questions of the theory of perturbations of the original system of equations.