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In this work the experimental effect of a slow decay of the photoluminescence is studied theoretically in the case of quantum dots with an indirect energy band gap. The slow decay of the photoluminescence is considered as decay in time of the luminescence intensity, following the excitation of the quantum dot sample electronic system by a short optical pulse. In the presented theoretical treatment the process is studied as a single dot property. The inter-valley deformation potential interaction of the excited conduction band electrons with lattice vibrations is considered in the self-consistent Born approximation to the electronic self-energy. The theory is built on the non-equilibrium electronic quantum transport theory. The time dependence of the photoluminescence decay is estimated upon using a simple effective mass model. The numerical calculation of the considered model shows the power-law time characteristics of the photoluminescence decay in the long-time limit of the decay. We demonstrate that the nonadiabatic influence of the interaction of the conduction band electrons with the lattice vibrations provides a mechanism giving us the power-law time dependence of the photoluminescence intensity signal. This theoretical result emphasizes the role of the electron-phonon interaction in the nanostructures.