Remark on global existence of solutions to the 1D compressible Euler equation with time-dependent damping

Abstract: In this paper, we consider the 1D compressible Euler equation with the damping coefficient $\lambda/(1+t)^{\mu}$. Under the assumption that $0\leq \mu <1$ and $\lambda >0$ or $\mu=1$ and $\lambda > 2$, we prove that solutions exist globally in time, if initial data are small $C^1$ perturbation near constant states. In particular, we remove the conditions on the limit $\lim_{|x| \rightarrow \infty} (u (0,x), v (0,x))$, assumed in previous results.