Long Tails in the Long-Time Asymptotics of Quasi-Linear Hyperbolic-Parabolic Systems of Conservation Laws

The long-time behavior of solutions of systems of conservation laws has been extensively studied. In particular, Liu and Zeng [Mem. Amer. Math. Soc., 125 (1997), pp. viii–120] have given a detailed exposition of the leading order asymptotics of solutions close to a constant background state. In this paper, we extend the analysis of Liu and Zeng by examining higher order terms in the asymptotics in the framework of the so-called two-dimensional p-system, though we believe that our methods and results also apply to more general systems. We give a constructive procedure for obtaining these terms, and we show that their structure is determined by the interplay of the parabolic and hyperbolic parts of the problem. In particular, we prove that the corresponding solutions develop long tails.