The existence of a (unique) solution of the second order semilinear elliptic equation $$ \sum^{n}_{i,j=0}a_{ij}(x)u_{x_{i}x_{j}}+f(\nabla u,u,x)=0 $$ with $x=(x_{0},x_{1},\dots, x_{n})\in (s_{0},\infty )\times \Omega '$, for a bounded domain $\Omega '$, together with the additional conditions $$ \begin{array}{l} u(x)=0\quad \mbox{for } (x_{1},x_{2},\dots, x_{n})\in\partial \Omega '\\ \\ u(x)=\varphi (x_{1},x_{2},\dots, x_{n})\quad \mbox{for } x_{0}=s_{0}\\ \\ \vert u(x)\vert\quad\mbox{globally bounded} \end{array} $$ is shown to be a well posed problem under some sign and growth restrictions on $f$ and its partial derivatives. It can be seen as an initial value problem, with initial value $\varphi $, in the space ${\cal C}^{0}_{0}(\overline {\Omega '})$ and satisfying the strong order-preserving property. In the case that $a_{ij}$ and $f$ do not depend on $x_{0}$ or are periodic in $x_{0}$ it is shown that the corresponding dynamical system has a compact global attractor. Also, conditions on $f$ are given under which all the solutions tend to zero as $x_{0}$ tends to infinity. Proofs are strongly based on maximum and comparison techniques.