Consider a first order PDE of the following form: $$\frac{\partial u}{\partial t} = -\omega\frac{\partial u}{\partial x} + f(x,t,u,u_x ),$$ $$u(x, 0) = u_0,$$ where $u(x,t): \mathbb{\mathbb{T}}\times[0, 2\pi] \to \mathbb{C}^4$, $f$ is analytic, $u_0$ holomorphic on the strip $|\text{Im}(x)|<\sigma$, $\omega$ is a constant satisfying diophantine condtions:
there exist $\mu>0, \tau>1$ such that for any $m , n \in\mathbb{N}_+$, $$\left|\omega-\frac{m}{n}\right|>\mu|n|^{-\tau},$$ Are there any global results for the existence of holomorphic solutions with respect to $x$ to this PDE?
(In the original post, I didn't specify the solutions are holomorphic with respect to $x$.)