Summability in a monomial for some classes of singularly perturbed partial differential equation

TL;DR

This paper advances the understanding of asymptotic expansions and summability in monomials for multivariable singularly perturbed PDEs, introducing new characterizations and Tauberian theorems, and applying Borel-Laplace analysis to establish solution summability.

Contribution

It provides a novel characterization of monomial summability via bounded derivatives and develops Tauberian theorems, extending summability theory to singularly perturbed PDEs.

Findings

Characterization of asymptotic expansions in terms of bounded derivatives

Development of Tauberian theorems for summability processes

Proof of monomial summability for solutions of certain singularly perturbed PDEs

Abstract

The aim of this paper is to continue the study of asymptotic expansions and summability in a monomial in any number of variables. In particular we characterize these expansions in terms of bounded derivatives and we develop tauberian theorems for the summability processes involved. Furthermore, we develop and apply the Borel-Laplace analysis in this framework to prove the monomial summability of solutions of a specific class of singularly perturbed PDEs.