Asymptotic Behavior of Global Classical Solutions to the Mixed Initial-Boundary Value Problem for Quasilinear Hyperbolic Systems with Small BV Data

2009 ◽  
Vol 98 (1) ◽  
pp. 25-64 ◽  
Author(s):  
Zhi-Qiang Shao
Keyword(s):  
Asymptotic Behavior ◽  
Boundary Value Problem ◽  
Hyperbolic Systems ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Classical Solutions ◽  
Quasilinear Hyperbolic Systems ◽  
Global Classical Solutions ◽  
Initial Boundary Value

GLOBAL WEAKLY DISCONTINUOUS SOLUTIONS TO THE MIXED INITIAL–BOUNDARY VALUE PROBLEM FOR QUASILINEAR HYPERBOLIC SYSTEMS

2009 ◽  
Vol 19 (07) ◽  
pp. 1099-1138 ◽  
Author(s):  
ZHI-QIANG SHAO
Keyword(s):  
Boundary Value Problem ◽  
A Priori Estimates ◽  
Hyperbolic Systems ◽  
Boundary Value ◽  
Local Existence ◽  
Discontinuous Solution ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Quasilinear Hyperbolic Systems ◽  
Initial Boundary Value

In this paper, we consider the mixed initial–boundary value problem for first-order quasilinear hyperbolic systems with general nonlinear boundary conditions in the half space {(t, x) | t ≥ 0, x ≥ 0}. Based on the fundamental local existence results and global-in-time a priori estimates, we prove the global existence of a unique weakly discontinuous solution u = u(t, x) with small and decaying initial data, provided that each characteristic with positive velocity is weakly linearly degenerate. Some applications to quasilinear hyperbolic systems arising in physics and other disciplines, particularly to the system describing the motion of the relativistic closed string in the Minkowski space R1+n, are also given.

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