On the global existence of smooth solutions to the multi-dimensional compressible euler equations with time-depending damping in half space

This paper is a continue work of [4,5]. In the previous two papers, we studied the Cauchy problem of the multi-dimensional compressible Euler equations with time-depending damping term -μ(1+t)λρu, where λ≥0 and μ > 0 are constants. We have showed that, for all λ≥0 and μ>0, the smooth solution to the Cauchy problem exists globally or blows up in finite time. In the present paper, instead of the Cauchy problem we consider the initial-boundary value problem in the half space ℝd+ with space dimension d = 2,3. With the help of the special structure of the equations and the fluid vorticity, we overcome the difficulty arisen from the boundary effect. We prove that there exists a global smooth solution for 0 ≤ λ <1 when the initial data is close to its equilibrium state. In addition, exponential decay of the fluid vorticity will also be established.