On Parametric Gevrey Asymptotics for Singularly Perturbed Partial Differential Equations with Delays

Abstract

We study a family of singularly perturbed $q$-difference-differential equations in the complex domain. We provide sectorial holomorphic solutions in the perturbation parameter $ϵ$. Moreover, we achieve the existence of a common formal power series in $ϵ$ which represents each actual solution and establish $q$-Gevrey estimates involved in this representation. The proof of the main result rests on a new version of the so-called Malgrange-Sibuya theorem regarding $q$-Gevrey asymptotics. A particular Dirichlet like series is studied on the way.