The mixed initial-boundary value problem for diagonalizable quasilinear hyperbolic systems in the first quadrant

Abstract

In this paper, we investigate the mixed initial-boundary value problem for diagonalizable quasilinear hyperbolic systems with nonlinear boundary conditions on a half-unbounded domain { ( t , x ) | t ≥ 0 , x ≥ 0 } . Under the assumptions that system is strictly hyperbolic and linearly degenerate, we obtain the global existence and uniqueness of C 1 solutions with the bounded L 1 ∩ L ∞ norm of the initial data as well as their derivatives and appropriate boundary condition. Based on the existence results of global classical solutions, we also prove that when t tends to infinity, the solutions approach a combination of C 1 travelling wave solutions. Under the appropriate assumptions of initial and boundary data, the results can be applied to the equation of time-like extremal surface in Minkowski space R 1 + ( 1 + n ) .