Gevrey versus q-Gevrey asymptotic expansions for some linear q-difference-differential Cauchy problem

TL;DR

This paper investigates the asymptotic behavior of solutions to certain singularly perturbed q-difference-differential equations, revealing different Gevrey and q-Gevrey asymptotic expansions through norm modifications and advanced analytical techniques.

Contribution

It introduces new asymptotic expansions of Gevrey and mixed type for q-difference-differential equations, utilizing norm modifications and Ramis-Sibuya theorem applications.

Findings

Identifies Gevrey and q-Gevrey asymptotic expansions for solutions

Demonstrates the impact of norm modifications on asymptotic behavior

Applies advanced path deformation and Ramis-Sibuya theorem techniques

Abstract

The asymptotic behavior of the analytic solutions of a family of singularly perturbed q-difference-differential equations in the complex domain is studied. Different asymptotic expansions with respect to the perturbation parameter and to the time variable are provided: one of Gevrey nature, and another of mixed type Gevrey and q-Gevrey. This asymptotic phenomena is observed due to the modification of the norm established on the space of coefficients of the formal solution. The techniques used are based on the adequate path deformation of the difference of two analytic solutions, and the application of several versions of Ramis-Sibuya theorem