On parametric multilevel q-Gevrey asymptotics for some linear Cauchy problem

TL;DR

This paper develops a $q$-Gevrey asymptotic analysis for a linear $q$-difference-differential Cauchy problem involving a perturbation parameter, extending previous work through a $q$-analog of acceleration and Ramis-Sibuya theorems.

Contribution

It introduces a $q$-analog of the Ramis-Sibuya theorem with two $q$-Gevrey orders, generalizing prior results to more complex forcing terms and asymptotic behaviors.

Findings

Establishes a $q$-Gevrey asymptotic framework for the problem.

Provides a $q$-analog of the Ramis-Sibuya theorem with two orders.

Demonstrates application to a related forcing term problem.

Abstract

We study a linear $q-$difference-differential Cauchy problem, under the action of a perturbation parameter $\epsilon$. This work deals with a $q-$analog of the research made in a previoues work, giving rise to a generalization of a recent work by the second author. This generalization is related to the nature of the forcing term which suggests the use of a $q-$analog of an acceleration procedure. The proof leans on a $q-$analog of the so-called Ramis-Sibuya theorem which entails two distinct $q-$Gevrey orders. The work concludes with an application of the main result when the forcing term solves a related problem.