Abstract: This paper is devoted to study on the asymptotic behaviors of solutions to a model of hyperbolic balance laws with damping on the quarter plane $(x,t)\in {\mathbb R}_{+}\times {\mathbb R}_{+}$. We show the optimal convergence rates of the solutions to their corresponding nonlinear diffusion waves, which are the solutions of the corresponding nonlinear parabolic equation given by the related Darcy's law. The optimal rates we obtained improve those in recent works on the IBVP by K. Nishihara and T. Yang [{\small \it J. Differential Equations} {\small \bf 156} (1999) 439-458] and by P. Marcati and M. Mei [{\small \it Quart. Appl. Math.} {\small \bf 56}(2000)]. The energy method with the method of Fourier transform together are efficiently used for proof.
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