This　paper　is　concerned　with　the　asymptotic　behavior　of　global　C1　solutions　of　the　Goursat　problem　for　quasilinear　hyperbolic　systems.　Based　on　the　existence　result　on　the　global　classical　solution,　we　prove　that　when　t　tends　to　the　infinity,　the　solution　approaches　a　combination　of　Lipschitz　continuous　and　piecewise　C1　traveling　wave　solutions,　provided　that　the　C1　norm　of　the　boundary　data　is　bounded　but　possibly　large,　and　the　BV　norm　of　the　boundary　data　is　sufficiently　small.　Applications　include　the　1D　compressible　Euler　equations　for　Chaplygin　gases.　©　2013　Springer　Basel.