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.

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.

(Erratum: This belief seems to be wrong. See Thomas Rot's comment below.)

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.

I don't know how to exclude the second possibility.

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.