Asymptotic Behavior of Solutions for p-System with Relaxation

AbstractIn this paper, we consider the asymptotic behavior of solutions for the Cauchy problem for p-system with relaxationvt−ux=0,ut+p(v)x=1ε(f(v)−u), (E) with initial data(v, u)(x, 0)=(v 0(x), u 0(x))→(v±, u±), v±>0,asx→± ∞. (I) We are interested to show the solutions of (E), (I) tend also to the equilibrium rarefaction waves and the traveling waves even if the limits (v±, u±) of the initial data at x=±∞ do not satisfy the equilibrium equation; i.e., u±≠f(v±). When the limits of the initial data at infinity satisfy equilibrium states, Liu [9] studied the stability of rarefaction waves and traveling waves for the general 2×2 hyperbolic conservation laws with relaxation