Asymptotic behavior of solutions to the compressible Euler equations with space-dependent damping on the half line

We investigate the large time behavior of solutions to the initial-boundary value problem of the one dimensional compressible Euler equations with space-dependent damping on the half line. It is shown that the unique solution of the compressible Euler equations with space-dependent damping globally exists and converges time-asymptotically to the diffusion wave at the optimal convergence rate, which coincides with the heat equation, and particularly improves the convergence rate of the recent work by Matsumura-Nishihara (SIAM J. Math. Anal., 56(2024), 993-1015). Compared to all the previous related works where either the damping coefficient is space-independent or the boundary effects are ignored, this gives the first optimal convergence rate for the compressible Euler equations with damping in the presence of both space-dependent damping and boundary effects. Our method of proof consists of the choice of new correction functions, the technical time-weighted energy estimates and the Green function method.