Solution of second-order hyperbolic quasilinear systems with spatio-characteristic initial data in weighted Sobolev-type spaces under finite differentiability assumptions on the data

TL;DR

This paper proves existence and uniqueness of solutions for second-order hyperbolic quasilinear systems with spatio-characteristic initial data in weighted Sobolev spaces, extending previous work and applying to Einstein equations.

Contribution

It establishes new semi-global existence and uniqueness results for Goursat problems with linear coefficient dependence in weighted Sobolev spaces.

Findings

Proves existence and uniqueness in weighted Sobolev spaces for hyperbolic systems.

Extends previous results to semi-global solutions for Goursat problems.

Applies results to harmonic gauge vacuum Einstein equations.

Abstract

The aim of this work is to establish an existence and uniqueness solution for spatiocharacteristic second-order quasilinear hyperbolic problems in Sobolev type spaces with weights to clarify and complete the previous work done by H. Muller Zum Hagen and H.J. Seifert, Gen. Rel. and Gravit. 1977. We use this result in P. G. Louokdom tamto, PhD thesis ongoing, 2026 to establish a semi-global existence and uniqueness result for second-order quasilinear Goursat problems where the coefficients of the second derivatives depend linearly on the unknown in weighted Sobolev-type spaces, which we will apply to the harmonic gauge vacuum Einstein equations