Gradient flow in simple settings

Question

I would like to solve an equation of the type $\partial_t X = \nabla U (X)$ in $\Omega \subset\mathbb{R}^2$ with an initial condition.

I know that under some conditions, $X(t)$ will converge to a critical point of $U$. I am looking for a reference where this issue is adressed and where the conditions of convergence are listed.

I know that if the critical point is not degenerate, it works. But in my settings, it seems I cannot prove that the critical point is non-degenerate, so I am looking for alternative conditions. $U$ is a solution of $-\Delta u = f$ with zero Dirichlet condition on $\partial \Omega$ and $f>0$ on $\Omega$.

So far I only found stuff for gradient flows in infinite-dimensional metric spaces (Ambrosio etc..). I only need results for finite dimension.

Thanks a lot for your help.

Answer 1

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.)