Large time behavior of solutions for nonlinear hyperbolic system with low-order dissipation (Q1288169)

The author deals with a system of hyperbolic conservation laws with damping modelling fluid flow in a pipe, which can be written in Lagrangian coordinates in the form (*) \(v_t-u_x=0\), \(u_t-p(v)_x +avu=0\). Large time behaviour of the solution of the Cauchy problem for (*) is investigated and it is shown that it tends to the solution of \(v_t-u_x=0\), \(p(v)_x+ avu=0\) with a rate of \(t^{-1/2}\). The initial function \(v_0\) for \(v\) is supposed to be sufficiently small but not only as a small perturbation of the constant state: the case \(\lim_{x\to-\infty} v_0(x)\neq \lim_{x\to+ \infty} v_0(x)\) is allowed.