Uniqueness of the solution of nonlinear singular first order partial differential equations

TL;DR

This paper proves the uniqueness of solutions for a class of nonlinear singular first-order PDEs with applications to the analytic continuation of holomorphic solutions.

Contribution

It establishes a weak-assumption-based proof of solution uniqueness for nonlinear singular PDEs and applies it to holomorphic solution continuation.

Findings

Uniqueness of solutions under weak conditions

Application to analytic continuation of holomorphic solutions

Extension of solution theory for singular PDEs

Abstract

This paper deals with nonlinear singular partial differential equations of the form $t \partial u/\partial t=F(t,x,u,\partial u/\partial x)$ with independent variables $(t,x) \in \mathbb{R} \times \mathbb{C}$, where $F(t,x,u,v)$ is a function continuous in $t$ and holomorphic in the other variables. Under a very weak assumption we show the uniqueness of the solution of this equation. The results are applied to the problem of analytic continuation of local holomorphic solutions of equations of this type.