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Open Access Research

Global existence and asymptotic behavior of classical solutions to Goursat problem for diagonalizable quasilinear hyperbolic system

Jianli Liu1 and Kejia Pan23*

Author Affiliations

1 Department of Mathematics, Shanghai University, Shanghai 200444, China

2 Key Laboratory of Metallogenic Prediction of Nonferrous Metals, Ministry of Education, School of Geosciences and Info-Physics, Central South University, Changsha 410083, China

3 School of Mathematics and Statistics, Central South University, Changsha 410075, China

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Boundary Value Problems 2012, 2012:36  doi:10.1186/1687-2770-2012-36

Published: 3 April 2012

Abstract

In this article, we investigate the global existence and asymptotic behavior of classical solutions to Goursat problem for diagonalizable quasilinear hyperbolic system. Under the assumptions that system is strictly hyperbolic and linearly degenerate, we obtain the global existence and uniqueness of C1 solutions with the bounded L1 ∩ Lnorm of the boundary data as well as their derivatives. Based on the existence result, we can prove that when t tends to in nity, the solutions approach a combination of piece-wised C1 traveling wave solutions. As the important example, we apply the results to the chaplygin gas system.

Mathematics Subject Classi cation (2000): 35B40; 35L50; 35Q72.

Keywords:
Goursat problem; global classical solutions; linearly degenerate; asymptotic behavior; traveling wave solutions.