Endowing $\mathbf{\Lambda}$ with a dynamic nature: constraints in a spatially curved Universe

In this study, we consider three dark energy models in which $\Lambda$ is not constant, but has a dynamic nature that depends on the Hubble parameter $H$ and/or its time derivative $\dot{H}$. We analyze the generalized running vacuum model, for which $\Lambda(H)=A+BH^2+C\dot{H}$, along with the two models obtained by setting $B$ or $C$ equal to zero. A null value for $C$ yields the classical running vacuum model (RVM), while $B=0$ corresponds to what we term the generalized running vacuum sub-case, or GRVS. Our main aim is to investigate whether these models can accommodate non-zero spatial curvature. To this end, we carry out a Markov Chain Monte Carlo analysis using data for the observables associated with Type-Ia supernovae, cosmic chronometers, the cosmic microwave background and baryon acoustic oscillations, as well as two values for the Hubble constant. Then we include data relating to the growth of large-scale structure (LSS) and repeat the procedure. Our results indicate that taking LSS observations into account helps to tighten constraints and determine a definite sign for the model parameters. In the case of the RVM and GRVS, the addition of growth data results in dynamical vacuum energy being preferred to a cosmological constant at a little over $1\sigma$. This happens in both the flat and non-flat scenarios -- there are only a few exceptions -- but comes at the cost of an extra parameter which can degrade the performance of the models (as assessed by model selection criteria). Of special relevance is the fact that the inclusion of LSS data appears to increase compatibility with a flat geometry. It also brings the constraints on the Hubble constant closer to the range of values established by \emph{Planck}.