Maximal domains of solutions for analytic quasilinear differential equations of first order

TL;DR

This paper investigates the maximal domain of analytic continuation for solutions to first-order quasi-linear PDEs, using characteristic vector fields and first integrals to determine the extension limits.

Contribution

It introduces a method to explicitly find the maximal domain of analytic solutions for first-order quasi-linear PDEs using characteristic methods and first integrals.

Findings

Maximal domains can be explicitly determined for certain PDEs.

The method applies to scalar conservation laws with global first integrals.

Examples demonstrate the applicability of the approach.

Abstract

We study the real-analytic continuation of local real-analytic solutions to the Cauchy problems of quasi-linear partial differential equations of first order for a scalar function. By making use of the first integrals of the characteristic vector field and the implicit function theorem we determine the maximal domain of the analytic extension of a local solution as a single-valued function. We present some examples including the scalar conservation laws that admit global first integrals so that our method is applicable.