Running vacuum model in a non-flat universe

We investigate observational constraints on the running vacuum model (RVM) of begin{document}$Lambda=3nu (H^{2}+K/a^2)+c_0$end{document} in a spatially curved universe, where begin{document}$nu$end{document} is the model parameter, begin{document}$K$end{document} corresponds to the spatial curvature constant, begin{document}$a$end{document} represents the scalar factor, and begin{document}$c_{0}$end{document} is a constant defined by the boundary conditions. We study the CMB power spectra with several sets of begin{document}$nu$end{document} and begin{document}$K$end{document} in the RVM. By fitting the cosmological data, we find that the best fitted begin{document}$chi^2$end{document} value for RVM is slightly smaller than that of begin{document}$Lambda$end{document}CDM in the non-flat universe, along with the constraints of begin{document}$nuleqslant O(10^{-4})$end{document} (68% C.L.) and begin{document}$|Omega_K=-K/(aH)^2|leqslant O(10^{-2})$end{document} (95% C.L.). In particular, our results favor the open universe in both begin{document}$Lambda$end{document}CDM and RVM. In addition, we show that the cosmological constraints of begin{document}$Sigma m_{nu}=0.256^{+0.224}_{-0.234}$end{document} (RVM) and begin{document}$Sigma m_{nu}=0.257^{+0.219}_{-0.234}$end{document} (begin{document}$Lambda$end{document}CDM) at 95% C.L. for the neutrino mass sum are relaxed in both models in the spatially curved universe.