Global existence and stability of temporal periodic solution to non-isentropic compressible Euler equations with a source term

In this paper, the 1-D compressible non-isentropic Euler equations with the source term $\beta \rho |u{|}^{\alpha }u$ in a bounded domain are considered. First, we study the existence of steady flows which can keep the upstream supersonic or subsonic state. Then, by wave decomposition and uniform prior estimations, we prove the global existence and stability of smooth solutions under small perturbations around the steady supersonic flow. Moreover, we get that the smooth supersonic solution is a temporal periodic solution with the same period as the boundary, after a certain start-up time, once the boundary conditions are temporal periodic.