Does there exist a global version of the Cauchy-Kowalevskaya theorem for linear PDEs ? Global in the sense that the solution exist and is analytic on $\mathbb{R}$ if so are the coefficients of the PDE. For reminder, the Cauchy-Kowalevskaya theorem only gives the local existence (under some conditions).
Example: $\partial_x u (x,y) +\partial_y u (x,y)=0$ and $u(0,y)=f(y)$, where $(x,y) \in \mathbb{R}^2$. Clearly, if $f$ is real analytic on $\mathbb{R}$ then $u$ is real analytic on $\mathbb{R}^2$. How to apply the Cauchy-Kowalevskaya theorem to recover this result ?