Convergence to nonlinear diffusion waves for solutions of the initial boundary problem to the hyperbolic conservation laws with damping
In this paper we consider a model of hyperbolic balance laws with damping on the quarter plane (x, t) e R+ x R+. By means of a suitable shift function, which will play a key role to overcome the difficulty of large boundary perturbations, we show that the IBVP solutions converge time-asymptotically to the shifted nonlinear diffusion wave solutions of the Cauchy problem to the nonlinear parabolic equation given by the related Darcy's law. We obtain also the time decay rates, which are the optimal ones in the L2-sense. Our proof is based on the use of the classical energy method.