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 $0 \leq \mu \lt 1$ and $\lambda \gt 0$ or $\mu = 1$ and $\lambda \gt 2$, 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.