Exponential type Nagumo norms and summability of formal solutions of singular partial differential equations

TL;DR

This paper develops a method using exponential Nagumo norms to analyze the summability and asymptotic behavior of formal solutions to a class of nonlinear degenerated singular PDEs, providing conditions for summability and constructing analytical solutions.

Contribution

It introduces a novel application of exponential Nagumo norms to extend Gevrey asymptotic analysis to holomorphic parameters and establishes sharp conditions for $k$-summability of formal solutions.

Findings

Established sharp $k$-summability conditions for formal solutions.

Extended Gevrey asymptotic analysis to holomorphic parameters.

Constructed analytical solutions in conical domains.

Abstract

In this paper, we study a class of first order nonlinear degenerated partial differential equations with singularity at $(t,x)=(0,0)\in \CC^2$. By means of exponential type Nagumo norm approach, Gevrey asymptotic analysis extends to case of holomorphic parameters by a natural way. A sharp condition is then established to deduce the $k$-summability for the formal solutions. Furthermore, analytical solutions in conical domains are found for each type of these nonlinear singular PDEs.