Asymptotics towards nonlinear diffusion waves for the solutions of a hyperbolic system with linear damping on quadrant

This article explores the asymptotic behaviour on the quarter plane $(x,t)\in R^{+}\times R^{+}$ of solutions of $M_1$ model. A more general system is considered here for the analysis. The global existence of solutions to the initial boundary value problem is first established under the constraints of small initial data and perturbations, which subsequently converge to their respective nonlinear diffusion waves, i.e., the solutions of the associated nonlinear parabolic equation arising from Darcy's law. Additionally, optimal convergence rates are established. The methodology employed relies on the energy method in conjunction with the Green's function method.