In this paper, the one-side physical vacuum problem for the one dimensional compressible Euler equations with time-dependent damping is considered. Near the physical vacuum boundary, the sound speed is $C^{1/2}$-Hölder continuous. The coefficient of the time-dependent damping is given by $\frac{\mu }{{(1+t)}^{\lambda }}$, $(0<\lambda ,0<\mu )$ which decays by order −λ in time. First we give an one-side physical vacuum background solution whose density and velocity have a growing order with respect to time. Then the main purpose of this paper is to prove the stability of this background solution under the assumption that $0<\lambda <1,0<\mu$ or $\lambda =1,2<\mu$. The pointwise convergence rate of the density, velocity and the expanding rate of the physical vacuum boundary are also given. The proof is based on the space-time weighted energy estimates, elliptic estimates and the Hardy inequality in the Lagrangian coordinates.

本文考虑了一类具有时变阻尼的一维可压缩欧拉方程的单侧物理真空问题。在物理真空边界附近，声速为$C^{\mathrm{1个}/2}$-霍尔德连续。随时间变化的阻尼系数由下式给出$\frac{\mu }{{（\mathrm{1个}+Ť）}^{\lambda }}$， $（0<\lambda ，0<\mu ）$它随时间衰减-λ。首先，我们给出了一侧物理真空背景溶液，该溶液的密度和速度随时间增长。那么，本文的主要目的是在以下假设下证明该背景解的稳定性：$0<\lambda <\mathrm{1个}，0<\mu$ 要么 $\lambda =\mathrm{1个}，2<\mu$。还给出了物理真空边界的密度，速度和扩展率的逐点收敛速度。该证明基于时空加权能量估计，椭圆估计和拉格朗日坐标中的Hardy不等式。