On parametric Gevrey asymptotics for singularly perturbed partial differential equations with delays

TL;DR

This paper investigates $q$-difference-differential equations with delays, establishing sectorial solutions, formal power series representations, and $q$-Gevrey asymptotics using a novel Malgrange-Sibuya theorem extension.

Contribution

It introduces a new version of the Malgrange-Sibuya theorem for $q$-Gevrey asymptotics and applies it to singularly perturbed equations with delays.

Findings

Existence of sectorial holomorphic solutions in the perturbation parameter.

Presence of a common formal power series representing all solutions.

Establishment of $q$-Gevrey estimates for the solutions.

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 $\epsilon$. Moreover, we achieve the existence of a common formal power series in $\epsilon$ 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.