Propagation of analyticity for a class of nonlinear hyperbolic equations

TL;DR

This paper proves that solutions to a specific class of nonlinear weakly hyperbolic equations in one space variable preserve their analyticity over time if initially analytic, under certain boundedness conditions.

Contribution

It establishes the propagation of analyticity for solutions of a particular class of nonlinear hyperbolic equations with time-dependent coefficients in one spatial dimension.

Findings

Analyticity persists over time for solutions with initially analytic data.

The result applies to bounded solutions in C-infinity for a class of homogeneous equations.

Time-dependent coefficients do not destroy initial analyticity.

Abstract

The propagation of analyticity for a solution u(t,x) to a nonlinear weakly hyperbolic equation of order m, means that if u, and its time derivatives up to the order m-1, are analytic in the space variables x at the initial time, then they remain analytic for any time. Here we prove that such a property holds for the solutions bounded in C-infinity of a special class of homogeneous equations in one space variable, with time dependent coefficient.