ASYMPTOTIC DECAY TOWARD RAREFACTION WAVE FOR A HYPERBOLIC-ELLIPTIC COUPLED SYSTEM ON HALF SPACE

摘要

We consider the asymptotic behavior of solutions to a model of hyperbolicelliptic coupled system on the half-line R+=(0,∞),ut+uux+qx=0, -qxx+q+ux=0,with the Dirichlet boundary condition u(0,t)=0.S.Kawashima and Y.Tanaka [Kyushu J.Math.,58(2004),211-250]have shown that the solution to the corresponding Cauchy problem behaviors like rarefaction waves and obtained its convergence rate when u_＜u+.Our main concern in this paper is the boundary effect.In the case of null-Dirichlet boundary condition on u,asymptotic behavior of the solution(u,q)is proved to be rarefaction wave as t tends to infinity.Its convergence rate is also obtained by the standard L2-energy method and L1-estimate.It decays much lower than that of the corresponding Cauchy problem.