Suppose that $G$ is a linear, compact and connected Lie group and that we have a map $$ u:(0,\infty) \times G \longrightarrow \mathbb{C}$$
Also we suppose that for all $t_{0}\in(0,\infty)$ the map $u(t_{0},\cdot)$ is in $\mathcal{C}^{\infty}(G,\mathbb{C})$ and for all $x_{0}\in G$ the map $u(\cdot , x_{0})$ is $C^{\infty}((0,\infty), \mathbb{C})$.
This implies that $u$ is in $\mathcal{C}^{\infty}((0 , \infty) \times G$)? If the answer is no, what additional conditions do I need to have $u$ is in $\mathcal{C}^{\infty}((0 , \infty) \times G$)?