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We study a nonlinear initial value Cauchy problem depending upon a complexperturbation parameter $\epsilon$ whose coefficients depend holomorphically on$(\epsilon,t)$ near the origin in $\mathbb{C}^{2}$ and are bounded holomorphicon some horizontal strip in $\mathbb{C}$ w.r.t the space variable. We considera 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 ofsome neighborhood of 0 in $\mathbb{C}^{\ast}$, which are asked to share acommon formal power series asymptotic expansion of some Gevrey order as$\epsilon$ tends to 0. The proof leans on a version of the so-calledRamis-Sibuya theorem which entails two distinct Gevrey orders. Finally, we givean application to the study of parametric multi-level Gevrey solutions for somenonlinear initial value Cauchy problems with holomorphic coefficients andforcing term in $(\epsilon,t)$ near 0 and bounded holomorphic on a strip in thecomplex space variable.

On parametric multisummable formal solutions to some nonlinear initial value Cauchy problems