Abstract
Hsiao and Serre in [Chinese Ann. Math. Ser. B, 16B (1995), pp. 1--14] showed the solution to the system $$ \left\{\begin{array}{@{}ll} v_t-u_x = 0, & (t,x) \in R_+ \times R, \\[3pt] u_t+p(v,s)_x =-\alpha u, &\alpha > 0, \\[3pt] s_t = 0 \end{array}\right. $$ with initial data $$ (v,u,s)(0,x) = (v_0,u_0,s_0)(x) \rightarrow (\underline{v},u_{\pm}, \underline{s})\qquad \text{ as } x \rightarrow \pm\infty $$ tends to the following nonlinear parabolic equation time-asymptotically: $$ \left\{\begin{array}{@{}ll} \tilde{v}_t = -\frac 1{\alpha} p(\tilde{v},s_0)_{xx}, & (t,x) \in R_+ \times R, \\[3pt] \tilde{u} = -\frac 1{\alpha} p(\tilde{v},s_0)_{x}. \end{array}\right. $$ In this paper we find its convergence rate, which will be optimal.
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