Long-time behavior of solutions to the $ M_1 $ model with boundary effect

Abstract

In this paper, we are concerned with the asymptotic behavior of solutions of $ M_1 $ model on quadrant $ (x,t) \in \mathbb{R}^{+} \times \mathbb{R}^{+} $. From this model, combined with damped compressible Euler equations, a more general system is introduced. We show that the solutions to the initial boundary value problem of this system globally exist and tend time-asymptotically to the corresponding nonlinear parabolic equation governed by the related Darcy's law. Compared with previous results on compressible Euler equations with damping obtained by Nishihara and Yang in [26], and Marcati, Mei and Rubino in [17], the better convergence rates are obtained. The approach adopted is based on the technical time-weighted energy estimates together with the Green's function method.