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Extending recent work to finite temperatures, we calculate the Landau dampingof a Bogoliubov excitation in an optical lattice, due to coupling to a thermalcloud of such excitations. For simplicity, we consider a 1D Bose-Hubbard modeland restrict ourselves to the first energy band. For energy conservation to besatisfied, the excitations in the collision processes must exhibit ``anomalousdispersion'', analogous to phonons in superfluid $^4\rm{He}$. This leads to thedisappearance of all damping processes when $U n^{\rm c 0}\ge 6t$, where $U$ isthe on-site interaction, $t$ is the hopping matrix element and $n^{\rm c 0}(T)$is the number of condensate atoms at a lattice site. This phenomenon alsooccurs in 2D and 3D optical lattices. The disappearance of Beliaev dampingabove a threshold wavevector is noted.Comment: 4pages, 5figures, submitted to Phys. Rev. Let

Damping of Bogoliubov excitations in optical lattices