  The main objective of this paper is to study the time decay of
transport-diffusion equation with inhomogeneous localized damping in the
multi-dimensional torus. The drift is governed by an autonomous Lipschitz
vector field and the diffusion by the standard heat equation with small
viscosity parameter $\nu$. In the first part we deal with the inviscid case and
show some results on the time decay of the energy using in a crucial way the
ergodicity and the unique ergodicity of the flow generated by the drift. In the
second part we analyze the same problem with small viscosity and provide quite
similar results on the exponential decay uniformly with respect to the
viscosity in some logarithmic time scaling of the \mbox{type $t\in
[0,C_0\ln(1/\nu)]$}.
