In this paper, we will show the blowup phenomenon of solutions to the compressible Euler equations with time-dependent damping. Firstly, under the assumptions that the radially symmetric initial data and initial density contains vacuum states, the singularity of the classical solutions will formed in finite time in $\mathbb{R}^n (n \geq 2)$. Furthermore, we can also find a sufficient condition for the functional of initial data such that smooth solution of the irrotational compressible Euler equations with time-dependent damping breaks down in finite time for all kinds of fractional coefficients in $\mathbb{R}^n (n \geq 2)$.

中文翻译：

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具有时变阻尼的可压缩欧拉方程的$ C ^ 1 $解的爆破。

在本文中，我们将展示带时变阻尼的可压缩Euler方程解的爆炸现象。首先，在径向对称的初始数据和初始密度包含真空状态的假设下，经典解的奇异性将在有限时间内以$ \ mathbb {R} ^ n（n \ geq 2）$的形式形成。此外，我们还可以为初始数据的功能找到充分的条件，使得对于$ \ mathbb {R} ^ n中的所有分数系数，具有时变阻尼的可旋转可压缩Euler方程的光滑解在有限时间内分解。 （n \ geq 2）$。