Large time behavior of entropy solutions to some hyperbolic system with dissipative structure

In this paper the large time behavior of the global L ^∞ entropy solutions to the hyperbolic system with dissipative structure is investigated. It is proved that as t → ∞ the entropy solutions tend to a constant equilibrium state in L ² norm with exponential decay even when the initial values are arbitrarily large. As an illustration, a class of 2 × 2 system is studied.