# On singularly perturbed linear initial value problems with mixed irregular and Fuchsian time singularities

## TL;DR

This paper studies a family of linear PDEs with complex perturbation parameters, combining irregular and Fuchsian singularities, and constructs sectorial solutions using multisummability, revealing their Gevrey asymptotic bounds.

## Contribution

It introduces a novel approach to construct sectorial solutions for PDEs with mixed singularities using multisummability techniques.

## Key findings

- Constructed sectorial solutions via Laplace and Fourier transforms.
- Established Gevrey bounds depending on the singularity types.
- Extended previous work to include Fuchsian operators in the analysis.

## Abstract

We consider a family of linear singularly perturbed PDE relying on a complex perturbation parameter $\epsilon$. As in a former study of the authors (A. Lastra, S. Malek, Parametric Gevrey asymptotics for some nonlinear initial value Cauchy problems, J. Differential Equations 259 (2015), no. 10, 5220--5270), our problem possesses an irregular singularity in time located at the origin but, in the present work, it entangles also differential operators of Fuchsian type acting on the time variable. As a new feature, a set of sectorial holomorphic solutions are built up through iterated Laplace transforms and Fourier inverse integrals following a classical multisummability procedure introduced by W. Balser. This construction has a direct issue on the Gevrey bounds of their asymptotic expansions w.r.t $\epsilon$ which are shown to bank on the order of the leading term which combines both irregular and Fuchsian types operators.