Analyticity of solutions for quasilinear wave equations and other quasilinear systems

TL;DR

This paper proves that solutions to quasilinear wave equations with analytic initial data remain analytic throughout their existence, and extends this result to other quasilinear systems, including partial space-analyticity cases.

Contribution

It establishes the persistence of analyticity for solutions of quasilinear wave equations and other systems, including partial space-analyticity, under broad conditions.

Findings

Analytic solutions persist as long as the classical solution exists.

Analyticity in part of the space variables is also preserved.

The approach applies to various quasilinear equations.

Abstract

We prove the persistence of analyticity for classical solution of the Cauchy problem for quasilinear wave equations with analytic data. Our results show that the analyticity of solutions, stated by the Cauchy-Kowalewski and Ovsiannikov-Nirenberg theorems, lasts till a classical solution exists. Moreover, they show that if the equation and the Cauchy data are analytic only in a part of space-variables, then a classical solution also is analytic in these variables. The approach applies to other quasilinear equations and implies the persistence of the space-analyticity (and the partial space-analyticity) of their classical solutions.