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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 start a magnetic ordering at the Curie temperature of 201.5(5)~K. Thenon-ferromagnetic phases mainly consist of antiferromagnetic-clusters with sizesmaller than the ferromagnetic-clusters. Below Curie temperature the systemexhibits an irreversibility in the field cooled and zero field cooledmagnetization and frequency dependence in the peak of ac susceptibility. Thesedynamical features indicate towards the possible coexistence of spin-glassphase along with ferromagnetic-clusters similar to La$_{1-x}$Sr$_{x}$CoO$_3$(x$\geq$0.18), but the absence of field divergence in third harmonic of acsusceptibility and zero field cooled memory clearly rule out any suchpossibility. We argue that the spin-glass phase in La$_{1-x}$Sr$_{x}$CoO$_3$(x$\geq$0.18) is associated with the presence of incommensurateantiferromagnetic ordering in non-ferromagnetic phases which is absent inLa$_{0.5}$Ba$_{0.5}$CoO$_3$. Our analysis show that the observed dynamicalfeatures in La$_{0.5}$Ba$_{0.5}$CoO$_3$ are possibly due to progressive thermalblocking of ferromagnetic-clusters which is further confirmed by theWohlfarth's model of superparamagnetism. The frequency dependence of the peakof ac susceptibility obeys the Vogel-Fulcher law with $\tau_0\approx10^{-9}$s.This together with the existence of an AT line in H-T space indicates thepresence of significant inter-cluster interaction among theseferromagnetic-clusters.

Coexistence of interacting-ferromagnetic and small-antiferromagnetic clusters in La$_{0.5}$Ba$_{0.5}$CoO$_3$