Assume $D\subset \mathbb{R}^d$ is a domain,
$f:D\rightarrow \mathbb{R}$ smooth
$u:[0,\infty)\rightarrow D$ is vector valued and bounded, where $u=u(t)$
is satisfying $u'+\nabla_u f(u)=0$
Now, assume $\exists \{t_k\}\subset [0,\infty)$ such that $\lim_{k\to\infty} u(t_k)=y\in \mathbb{R}$
Is $y$ the unique limit point of $u$?
Apparently this question is harder to answer for $d\geq 2$, and you need the extra assumption that $f$ is analytic. My professor said for $d=1$ it is doable without and extra assumptions of $f$, where the equation becomes $u'+f'(u)=0$
Anyone have any knowledge on what the approach is?