On parametric Borel summability for linear singularly perturbed Cauchy problems with linear fractional transforms

TL;DR

This paper develops a framework for constructing holomorphic solutions to a class of singularly perturbed Cauchy problems involving fractional transforms, revealing complex Gevrey asymptotics and summability properties.

Contribution

It introduces a novel method combining Laplace transforms and Gevrey asymptotics to analyze solutions with fractional transforms in singular perturbation problems.

Findings

Construction of solutions with sectorial holomorphicity

Identification of two levels of Gevrey asymptotics

Unicity results for the $1^{+}$ asymptotic layer

Abstract

We consider a family of linear singularly perturbed Cauchy problems which combines partial differential operators and linear fractional transforms. We construct a collection of holomorphic solutions on a full covering by sectors of a neighborhood of the origin in $\mathbb{C}$ with respect to the perturbation parameter $\epsilon$. This set is built up through classical and special Laplace transforms along piecewise linear paths of functions which possess exponential or super exponential growth/decay on horizontal strips. A fine structure which entails two levels of Gevrey asymptotics of order 1 and so-called order $1^{+}$ is witnessed. Furthermore, unicity properties regarding the $1^{+}$ asymptotic layer are observed and follow from results on summability w.r.t a particular strongly regular sequence recently obtained in a previous study.