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

TL;DR

This paper investigates parametric multisummable formal solutions for nonlinear initial value problems with complex perturbation parameters, establishing their existence and properties using a Ramis-Sibuya theorem extension.

Contribution

It introduces a novel approach to analyze multisummable solutions with multiple Gevrey orders for nonlinear problems depending on complex parameters.

Findings

Existence of multisummable formal solutions with two Gevrey orders.

Application to parametric multi-level Gevrey solutions in nonlinear initial value problems.

Extension of Ramis-Sibuya theorem to handle multiple Gevrey orders.

Abstract

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.