Asymptotic Behavior Of Global Smooth Solutions For A Class Of Hyperbolic Conservation Laws With Relaxation

In this paper,we discuss the asymptotic behavior related to some basic strong nonlinear waves of global smooth solutions for a class of systems of hyperbolic conservation laws with relaxation under large initial disturbance. Our main results are made up of two parts. First,we show that the global smooth solutions of the p-system with relaxation converges to the correspond- ing nonlinear rarefaction waves for the case when the corresponding Riemann problem of hyperbolic conservation laws admits a rarefaction wave solution and in our analysis ,the initial error can be chosen arbitrarily large. Second, we obtain the convergence rates to the travelling waves of the global smooth solutions for a semi-linear system with relaxation of hyperbolic conservation laws without any smallness restriction to the initial disturbance . The two results stated as above have showed some improvement compared with the former results in this direction.The first chapter are concerned with the large time behavior of global smooth solutions to the Cauchy problem of the following p-system with re- laxation. with initial data Here v+＞v＞0 and p(v) and f(v) are smooth function ofv satisfying p′(v)＜0 and f″(v)＜0 for v＞0 . Furthermore, we assume that they satisfy the following sub-characteristic condition Former results in this direction indict that such a problem has a global smooth solution provided that the first derivative of the solutions with re- spect to the space variable x are sufficiently small. Under the same small assumption of on the global smooth solution, we show that it converges to the corresponding nonlinear rarefaction wave and in our analysis, we do not ask the rarefaction wave to be weak and the initial error can also be chosen arbitrarily large.In the second chapter, we are concerned with the convergence rates to travelling waves for a relaxation model with general flux functions as follows with initial data (u(x,O), v(x, O)) =(u_o(x),u_o(x)) where (U_o, V_o)(X)→(u±, v±) as x→±∞, with flux functions f(u)∈C2(R) and satisfy the following sub-characteristic condition Here a > 0 is a fixed constant,O <ε<< 1 is the relaxation time. Compared with former results in this direction ,the main novelty in this paper lies in the fact that the initial disturbance can be chosen large in suitable norm.Keywords: Convergence rate, relaxation, traveling waves, large initial disturbance, Strong nonlinear rarefaction waves, p—system with relaxation...