Summary: In this paper, we are concerned with the global existence and blowup of smooth solutions to the multi-dimensional compressible Euler equations with time-depending damping
\[ \begin{cases} \partial_t\rho+\operatorname{div}(\rho u)=0,\\ \partial_t(\rho u)+\operatorname{div} (\rho u\otimes u + p\mathrm{I}_d)=-\alpha(t)\rho u),\\ \rho(0,x)=\overline{\rho}+\varepsilon\rho_0(x), \quad u(0,x)=\varepsilon u_0(x), \end{cases} \]
where \(x=(x_1, \cdots, x_d)\in\mathbb R^d\) \((d=2,3)\), the frictional coefficient is \(\alpha(t)=\frac{\mu}{(1+t)^\lambda}\) with \(\lambda\geqslant0\) and \(\mu>0\), \(\overline{\rho}>0\) is a constant, \(\rho_0, u_0 \in C_0^\infty({\mathbb R}^d)\) , \((\rho_0, u_0)\not\equiv 0\), \(\rho(0, x)>0\), and \(\varepsilon>0\) is sufficiently small. One can totally divide the range of \(\lambda\geqslant0\) and \(\mu>0\) into the following four cases:

Case

\(0\leqslant\lambda<1\) , \(\mu>0\) for \(d=2, 3\);

Case

\(\lambda=1\) , \(\mu>3-d\) for \(d=2, 3\);

Case

\(\lambda=1\) , \(\mu\leqslant 3-d\) for \(d=2\);

Case

\(\lambda>1\) , \(\mu>0\) for \(d=2, 3\).

We show that there exists a global \(C^{\infty}\)-smooth solution \((\rho, u)\) in Case 1, and Case 2 with \(\operatorname{curl} u_0\equiv 0\), while in Case 3 and Case 4, for some classes of \((\rho_0(x), u_0(x))\), the solution \((\rho, u)\) will blow up in finite time. Therefore, \(\lambda=1\) and \(\mu=3-d\) appear to be the critical power and critical value, respectively, for the global existence of small amplitude smooth solution \((\rho, u)\) in \(d\)-dimensional compressible Euler equations with time-depending damping.