Global existence of analytic solutions to the Cauchy problem in a complex domain (Q2472527)

Let \(\Omega\) be a complex domain and \(f:\Omega \times \mathbb C^n \to \mathbb C^n\) be an analytic map. The author discusses the Cauchy problem for the following system of differential equations \[ \begin{aligned} & u'(z) = f(z,u(z)),\quad z \in \Omega , \cr & u(0) = u_0 \in \mathbb C^n. \end{aligned} \] She establishes the existence of an analytic solution to the above system on a starshaped domain without assuming any growth condition on \(f\), if a so-called solution-tube exists for the system.