  We report detailed dc magnetization, linear and non-linear ac susceptibility
measurements on the hole doped disordered cobaltite
La$_{0.5}$Ba$_{0.5}$CoO$_3$. Our results show that the magnetically ordered
state of the system consists of coexisting non-ferromagnetic phases along with
percolating ferromagnetic-clusters. The percolating ferromagnetic-clusters
possibly start a magnetic ordering at the Curie temperature of 201.5(5)~K. The
non-ferromagnetic phases mainly consist of antiferromagnetic-clusters with size
smaller than the ferromagnetic-clusters. Below Curie temperature the system
exhibits an irreversibility in the field cooled and zero field cooled
magnetization and frequency dependence in the peak of ac susceptibility. These
dynamical features indicate towards the possible coexistence of spin-glass
phase along with ferromagnetic-clusters similar to La$_{1-x}$Sr$_{x}$CoO$_3$
(x$\geq$0.18), but the absence of field divergence in third harmonic of ac
susceptibility and zero field cooled memory clearly rule out any such
possibility. We argue that the spin-glass phase in La$_{1-x}$Sr$_{x}$CoO$_3$
(x$\geq$0.18) is associated with the presence of incommensurate
antiferromagnetic ordering in non-ferromagnetic phases which is absent in
La$_{0.5}$Ba$_{0.5}$CoO$_3$. Our analysis show that the observed dynamical
features in La$_{0.5}$Ba$_{0.5}$CoO$_3$ are possibly due to progressive thermal
blocking of ferromagnetic-clusters which is further confirmed by the
Wohlfarth's model of superparamagnetism. The frequency dependence of the peak
of 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 the
presence of significant inter-cluster interaction among these
ferromagnetic-clusters.
