  We study a nonlinear initial value Cauchy problem depending upon a complex
perturbation parameter $\epsilon$ whose coefficients depend holomorphically on
$(\epsilon,t)$ near the origin in $\mathbb{C}^{2}$ and are bounded holomorphic
on some horizontal strip in $\mathbb{C}$ w.r.t the space variable. We consider
a family of forcing terms that are holomorphic on a common sector in time $t$
and on sectors w.r.t the parameter $\epsilon$ whose union form a covering of
some neighborhood of 0 in $\mathbb{C}^{\ast}$, which are asked to share a
common formal power series asymptotic expansion of some Gevrey order as
$\epsilon$ tends to 0. The proof leans on a version of the so-called
Ramis-Sibuya theorem which entails two distinct Gevrey orders. Finally, we give
an application to the study of parametric multi-level Gevrey solutions for some
nonlinear initial value Cauchy problems with holomorphic coefficients and
forcing term in $(\epsilon,t)$ near 0 and bounded holomorphic on a strip in the
complex space variable.
