Multiscale Gevrey asymptotics in boundary layer expansions for some initial value problem with merging turning points

TL;DR

This paper develops multiscale Gevrey asymptotic expansions for solutions of a nonlinear singularly perturbed PDE with merging turning points, revealing different Gevrey orders in outer and inner solutions.

Contribution

It introduces a novel multiscale Gevrey asymptotic framework for boundary layer expansions in PDEs with complex perturbation parameters and merging turning points.

Findings

Outer and inner solutions have Gevrey asymptotic expansions.

Gevrey orders of solutions are generally different.

Constructs sectorial solutions in complex domains.

Abstract

We consider a nonlinear singularly perturbed PDE leaning on a complex perturbation parameter $\epsilon$. The problem possesses an irregular singularity in time at the origin and involves a set of so-called moving turning points merging to 0 with $\epsilon$. We construct outer solutions for time located in complex sectors that are kept away from the origin at a distance equivalent to a positive power of $|\epsilon|$ and we build up a related family of sectorial holomorphic inner solutions for small time inside some boundary layer. We show that both outer and inner solutions have Gevrey asymptotic expansions as $\epsilon$ tends to 0 on appropriate sets of sectors that cover a neighborhood of the origin in $\mathbb{C}^{\ast}$. We observe that their Gevrey orders are distinct in general.