Global existence  of classical solutions  of Goursat problem forquasilinear hyperbolic systems of diagonal form with large BV data

Zhi-Qiang Shao

doi:10.3934/cpaa.2013.12.2739

Article Contents

2013, Volume 12, Issue 6: 2739-2752. Doi: 10.3934/cpaa.2013.12.2739

This issue Previous Article Positive solutions of integral systems involving Bessel potentials Next Article On general fractional abstract Cauchy problem

Global existence of classical solutions of Goursat problem for quasilinear hyperbolic systems of diagonal form with large BV data

Zhi-Qiang Shao^1,

1.
Department of Mathematics, Fuzhou University, Fuzhou 350002

Received: October 2012

Revised: December 2012

Published: November 2013

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Abstract

We investigate the existence of a global classical solution to the Goursat problem for linearly degenerate quasilinear hyperbolic systems of diagonal form. As the result in [A. Bressan, Contractive metrics for nonlinear hyperbolic systems, Indiana Univ. Math. J. 37 (1988) 409-421] suggests that one may achieve global smoothness even if the $C^1$ norm of the initial data is large, we prove that, if the $C^1$ norm and the BV norm of the boundary data are bounded but possibly large, then the solution remains $C^1$ globally in time. Applications include the equation of time-like extremal surfaces in Minkowski space $R^{1+(1+n)}$ and the one-dimensional Chaplygin gas equations.

Keywords:

Quasilinear hyperbolic system of diagonal form,
Goursat problem,
global classical solution,
linear degeneracy,
large BV data,
Chaplygin gas equations.

Mathematics Subject Classification: Primary: 35L45, 35L60; Secondary: 35Q40.

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