Asymptotic behavior of solutions to the compressible Navier-Stokes
equation around a given constant state is investigated on
a cylindrical domain in $\mathbf{R}^{3}$, under the no slip
boundary condition for the velocity field. The $L^{2}$ decay
estimate is established for the perturbation from the constant
state. It is also shown that the time-asymptotic leading part
of the perturbation is given by a function satisfying a 1
dimensional heat equation. The proof is based on an energy
method and asymptotic analysis for the associated linearized
semigroup.