  This paper investigates the decay rates of the contact wave in
one-dimensional Navier-Stokes equations. We study two cases of perturbations,
with and without zero mass condition, i.e., the integration of initial
perturbations is zero and non-zero, respectively. For the case without zero
mass condition, we obtain the optimal decay rate $(1+t)^{-\frac{1}{2}}$ for the
perturbation in $L^\infty$ norm, which provides a positive answer to the
conjecture in \cite{HMX}. We applied the anti-derivative method, introducing
the diffusion wave to carry the initial excess mass, diagonalizing the
integrated system, and estimating the energy of perturbation in the
diagonalized system. Precisely, due to the presence of diffusion waves, the
decay rates for errors of perturbed system are too poor to get the optimal
decay rate. We find the dissipation structural in the diagonalized system, see
\cref{ds}. This observation makes us able to fully utilize the fact that the
sign of the derivative of the contact wave is invariant and to control the
terms with poor decay rates in energy estimates. For the case with zero mass
condition, there are also terms with poor decay rates. In this case, note that
there is a cancellation in the linearly degenerate field so that the terms with
poor decay rates will not appear in the second equation of the diagonalized
system. Thanks to this cancellation and a Poincar\'e type of estimate obtained
by a critical inequality introduced by \cite{HLM}, we get the decay rate of
$\ln^{\frac{1}{2}} (2+t)$ for $L^2$ norm of anti-derivatives of perturbation
and $(1+t)^{-\frac{1}{2}}\ln^{\frac{1}{2}}(2+t)$ for the $L^2$ norm of
perturbation itself, the decay rates are optimal, which is consistent with the
results obtained by using pointwise estimate in \cite{XZ} for the system with
artificial viscosity.
