Global existence of classical solutions to the mixed initial–boundary value problem for quasilinear hyperbolic systems of diagonal form with large BV data

In this paper, we investigate the mixed initial–boundary value problem with large BV data for linearly degenerate quasilinear hyperbolic systems of diagonal form with general nonlinear boundary conditions in the half space { ( t , x ) | t ⩾ 0 , x ⩾ 0 } . 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 initial and boundary data are bounded but possibly large, then the solution remains C 1 globally in time and possesses uniformly bounded total variation in x for all t ⩾ 0 . As an application, we apply the result to the system describing the motion of relativistic closed strings in the Minkowski space R 1 + n .