The maximum principle shows that $U>0$ inside $\Omega$. The function $U$ increases along the gradient flow lines, so if you start at a point $x_0$ inside $\Omega$, the trajectory $\Phi_t(p_0)$ $t>0$,  will stay forever inside.  The function $t\mapsto   U(\Phi_t(x_0))$  is strictly increasing so it has a limit 

$$ \lim_{t\to\infty} U(\Phi_t(x_0))=U_\infty\leq \max_{x\in\Omega} U(x). $$

The set $\omega_+(x_0)$ of limit points of the trajectory $\Phi_t(x_0)$, $t>0$ is thus  a subset of the level set $\{U(x)=U_\infty\}$. One the other hand, the set of limit points is  flow invariant.  So it it is a flow invariant set contained in a level set. The only gradient trajectories contained in a level set consist of stationary trajectories. The set $\omega_+(x_0)$ must therefore consist of  critical points of $U$.

If all the critical points of $U$ are *isolated* (they still could be degenerate),  then  the limit set must consist of single  critical point.  

More generally, if the limit set $\omega_+(x_0)$ contains an isolated critical point $p$, then a simple argument shows that $\omega_+(x_0)=\{p\}$. Thus $\omega_+(x_0)$  consists either of a single point, or all the points in $\omega_+(x_0)$ are non-isolated critical points.


Even in the second case, under some additional non-degeneracy assumptions, one can conclude  that a gradient trajectory has a unique limit point.  I believe this to be the case in general, but I don't have a proof.