  This paper is concerned with the multi-dimensional compressible Euler
equations with time-dependent damping of the form
$-\frac{\mu}{(1+t)^\lambda}\rho\boldsymbol u$ in $\mathbb R^n$, where $n\ge2$,
$\mu>0$, and $\lambda\in[0,1)$. When $\lambda>0$ is bigger, the damping effect
time-asymptotically gets weaker, which is called under-damping. We show the
optimal decay estimates of the solutions such that $\|\partial_x^\alpha
(\rho-1)\|_{L^2(\mathbb R^n)}\approx
(1+t)^{-\frac{1+\lambda}{2}(\frac{n}{2}+|\alpha|)}$, and $\|\partial_x^\alpha
\boldsymbol u\|_{L^2(\mathbb R^n)}\approx
(1+t)^{-\frac{1+\lambda}{2}(\frac{n}{2}+|\alpha|)-\frac{1-\lambda}{2}}$, and
see how the under-damping effect influences the structure of the Euler system.
Different from the traditional view that the stronger damping usually makes the
solutions decaying faster, here surprisingly we recognize that the weaker
damping with $0\le\lambda<1$ enhances the faster decay for the solutions. The
adopted approach is the technical Fourier analysis and the Green function
method. The main difficulties caused by the time-dependent damping lie in
twofold: non-commutativity of the Fourier transform of the linearized operator
precludes explicit expression of the fundamental solution; time-dependent
evolution implies that the Green matrix $G(t,s)$ is not translation invariant,
i.e., $G(t,s)\ne G(t-s,0)$. We formulate the exact decay behavior of the Green
matrices $G(t,s)$ with respect to $t$ and $s$ for both linear wave equations
and linear hyperbolic system, and finally derive the optimal decay rates for
the nonlinear Euler system.
