This work focuses on analyzing a spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW) universe within the Brans-Dicke (BD) scalar-tensor theory of gravity, incorporating a dynamical cosmological constant and Barrow holographic dark energy (BHDE) alongside pressureless dark matter (DM). The cosmological dynamics is explored by solving the field equations using an exponential form of the scale factor, and the scalar field evolving according to a power-law relationship. In this study, the Hubble horizon is chosen as the infrared (IR) cut-off, which plays a crucial role in defining the energy density of the BHDE. Both non-interacting and interacting scenarios between dark energy (DE) and dark matter (DM) are investigated to assess the influence of these components on cosmic evolution. To gain deeper insights, the analysis is further extended using cosmographic parameters, including the jerk, snap, and lerk parameters, as well as the Statefinder diagnostic pair r, s, which serve as powerful tools to characterize different phases of cosmic expansion. The findings reveal that the proposed model consistently replicates the accelerated expansion of the universe, in agreement with the current observational data. It is noteworthy that the dynamical behavior exhibited in the interacting scenario closely resembles that of the non-interacting case, thereby reinforcing consistency and the physical viability of the model.
Mahanta et al. (2026) studied this question.