  We study the damping of Bogoliubov excitations in an optical lattice at
finite temperatures. For simplicity, we consider a Bose-Hubbard tight-binding
model and limit our analysis to the lowest excitation band. We use the Popov
approximation to calculate the temperature dependence of the number of
condensate atoms $n^{\rm c 0}(T)$ in each lattice well. We calculate the Landau
damping of a Bogoliubov excitation in an optical lattice due to coupling to a
thermal cloud of excitations. While most of the paper concentrates on 1D
optical lattices, we also briefly present results for 2D and 3D lattices. For
energy conservation to be satisfied, we find that the excitations in the
collision process must exhibit anomalous dispersion ({\it i.e.} the excitation
energy must bend upward at low momentum), as also exhibited by phonons in
superfluid $^4\rm{He}$. This leads to the sudden disappearance of all damping
processes in $D$-dimensional simple cubic optical lattice when $U n^{\rm c
0}\ge 6DJ$, where $U$ is the on-site interaction, and $J$ is the hopping matrix
element. Beliaev damping in a 1D optical lattice is briefly discussed.
