Optimal Convergence Rates to Diffusion Waves for Solutions of the Hyperbolic Conservation Laws with Damping

Abstract.

This paper is devoted to study the asymptotic behaviors of the 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 L^p-rates $(1 + t)^{ - (1 - \tfrac{1} {{2p}})}$ for 2 ≤ p ≤ ∞ obtained in the present paper improve those $(1 + t)^{ - (\tfrac{3} {4} - \tfrac{1} {{2p}})}$ in the previous works on the IBVP by K. Nishihara and T. Yang [J. Differential Equations 156 (1999), 439–458] and by P. Marcati and M. Mei [Quart. Appl. Math. 56 (2000), 763–784]. Both the energy method and the method of Fourier transform are efficiently used to complete the proof.