Question about Cauchy-Kowalevsky theorem

Question

I have the following Cauchy problem:

$u_x+u_y=u^2$

$u(x,0)=h(x)$

The solution I got is: $u(x,y)=$$h(x-y)\over{1-yh(x-y)}$

If h is analytic, what can I conclude applying Cauchy-Kowalesky theorem? That there exists a unique analytic solution of the problem?

Thank you for your help.

Answer 1

Yes, there is a unique analytic solution in a (small) neighbourhood of the $x$-axis.