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In this paper, we consider the one-dimensional isentropic compressible Eulerequations with source term $\beta(t,x)\rho|u|^{\alpha}u$ in a bounded domain,which can be used to describe gas transmission in a nozzle.~The model isimposed a subsonic time-periodic boundary condition.~Our main results revealthat the time-periodic boundary can trigger an unique subsonic time-periodicsmooth solution and this unique periodic solution is stable under smallperturbations on initial and boundary data.~To get the existence of subsonictime-periodic solution, we use the linear iterative skill and transfer theboundary value problem into two initial value ones by using the hyperbolicproperty of the system. Then the corresponding linearized system can bedecoupled.~The uniqueness is a direct by-product of the stability. There is nosmall assumptions on the damping coefficient.

Global existence and stability of subsonic time-periodic solution to the damped compressible Euler equations in a bounded domain