On parametric Gevrey asymptotics for initial value problems with infinite order irregular singularity and linear fractional transforms

TL;DR

This paper extends previous work on Gevrey asymptotics for nonlinear PDEs by incorporating Moebius transforms and infinite order irregular operators, leading to new solutions with specific asymptotic properties.

Contribution

It introduces a novel regularization technique involving infinite order irregular operators combined with Moebius transforms in the PDE context.

Findings

Constructed solutions via Laplace and inverse Fourier transforms.

Derived Gevrey asymptotic expansions of order K>1.

Extended asymptotic analysis to PDEs with fractional transforms.

Abstract

This paper is a continuation a previous work of the authors where parametric Gevrey asymptotics for singularly perturbed nonlinear PDEs has been studied. Here, the partial differential operators are combined with particular Moebius transforms in the time variable. As a result, the leading term of the main problem needs to be regularized by means of a singularly perturbed infinite order formal irregular operator that allows us to construct a set of genuine solutions in the form of a Laplace transform in time and inverse Fourier transform in space. Furthermore, we obtain Gevrey asymptotic expansions for these solutions of some order $K>1$ in the perturbation parameter.