Asymptotic Decay Toward Rarefaction Wave for a Hyperbolic-elliptic Coupled System on Half Space

Abstract

" We consider the asymptotic behavior of solutions to a model of hyperbolic- elliptic coupled system on the half-line R_+ = (0,\u221e), u_t+uu_x+q_x=0, -q_{xx}+q+u_x=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 correspond- ing 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 L^2-energy method and L^1-estimate. It decays much lower than that of the corresponding Cauchy problem."