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We report detailed dc magnetization, linear and non-linear ac susceptibilitymeasurements on the hole doped disordered cobaltiteLa$_{0.5}$Ba$_{0.5}$CoO$_3$. Our results show that the magnetically orderedstate of the system consists of coexisting non-ferromagnetic phases along withpercolating ferromagnetic-clusters. The percolating ferromagnetic-clusterspossibly undergo a 3D Hisenberg like magnetic ordering at the Curie temperatureof 202(3) K. In between 202 and 220 K, the linear and non-linear acsusceptibility measurements show the presence of magnetic correlations evenwhen the spontaneous magnetization is zero which indicates the presence ofpreformed short range magnetic-clusters. The characteristics of these shortrange magnetic-clusters that exist above Curie temperature are quite distinctthan that of Griffiths phase e.g the inverse dc susceptibility exhibits anfield independent upward deviation, and the second harmonic of acsusceptibility is non-negative. Below Curie temperature the system exhibitspin-glass like features such as irreversibility in the field cooled and zerofield cooled magnetization and frequency dependence in the peak of acsusceptibility. The presence of a spin or cluster -glass like state is ruledout by the absence of field divergence in third harmonic of ac susceptibilityand zero field cooled memory. This indicates that the observed spin-glass likefeatures are possibility due to progressive thermal blocking offerromagnetic-clusters which is further confirmed by the Wohlfarth's model ofsuperparamagnetism. The frequency dependence of the peak of ac susceptibilityobeys the Vogel-Fulcher law with $\tau_0\approx 10^{-9}$ s. This together withthe existence of an AT line in H-T space indicates the existence of significantinter-cluster interaction among these ferromagnetic-clusters.

A critical examination of magnetic states of La$_{0.5}$Ba$_{0.5}$CoO$_3$: non-Griffiths phase and interacting ferromagnetic-clusters