Parametric formal Gevrey asymptotic expansions in two complex time variable problems

TL;DR

This paper develops a framework for Gevrey asymptotic expansions in two complex time variables for singularly perturbed PDEs, analyzing their analytic continuation and asymptotic behavior.

Contribution

It introduces a parametric approach to formal Gevrey asymptotics in two complex variables, addressing summability issues via Borel plane analysis.

Findings

Analytic continuation properties of solutions in the Borel plane are established.

Multiple exponential decay rates of solution differences are identified.

Asymptotic levels linking formal and analytic solutions are characterized.

Abstract

The analytic and formal solutions to a family of singularly perturbed partial differential equations in the complex domain involving two complex time variables are considered. The analytic continuation properties of the solution of an auxiliary problem in the Borel plane overcomes the absence of adequate domains which would guarantee summability of the formal solution. Moreover, several exponential decay rates of the difference of analytic solutions with respect to the perturbation parameter at the origin are observed, leading to several asymptotic levels relating the analytic and the formal solution.