I have 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 dimensionnal metric spaces (Ambrosio etc..). I only need results for finite dimension.

Thanks a lot for your help.

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.