Let $(M,g)$ be an n dimensional smooth Riemannian manifold, possibly noncompact. Suppose there exists a harmonic function $f$ defined on $M$, then we know that $f$ is smooth. Let $S$ be the set of critical points of $f$, i.e. $$ S=\{x\in M: |\nabla f (x)|=0\}. $$ For $x\in S$, we know that the gradient flow starting from $x$ remains unchanged.

For a point $x$ such that $|\nabla f(x)|\neq 0$, is it possible that the gradient flow reaches a critical point? If it's no, for a function $f$ defined on $M$ satisfying $\Delta f=c>0$, $c$ is a constant, do we have the same property?