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

We study a nonlinear initial value Cauchy problem depending upon a complex perturbation parameter $ε$ whose coefficients depend holomorphically on $(ε,t)$ 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 $ε$ 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 $ε$ 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 $(ε,t)$ near 0 and bounded holomorphic on a strip in the complex space variable.

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