Let $U\subset \mathbb{R}^2$ open, $\partial U\neq \varnothing$, $V\colon U\rightarrow \mathbb{R}^2$ smooth.
Let $c\colon [0,t_{max})\rightarrow U$ be an integral curve of $V$, where $t_{max}$ is the maximal time of existence (to the right). Assume $c(t)\xrightarrow{t\to t_{max}}c^*$ and $c^*\in\partial U$.
Are there any sufficent conditions for $t_{max}<\infty$?
(Can I conclude $t_{max}<\infty$ if I know that $V$ can not be extended continuously onto a neighborhood of $c^*$?)