  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 undergo a 3D Hisenberg like magnetic ordering at the Curie temperature
of 202(3) K. In between 202 and 220 K, the linear and non-linear ac
susceptibility measurements show the presence of magnetic correlations even
when the spontaneous magnetization is zero which indicates the presence of
preformed short range magnetic-clusters. The characteristics of these short
range magnetic-clusters that exist above Curie temperature are quite distinct
than that of Griffiths phase e.g the inverse dc susceptibility exhibits an
field independent upward deviation, and the second harmonic of ac
susceptibility is non-negative. Below Curie temperature the system exhibit
spin-glass like features such as irreversibility in the field cooled and zero
field cooled magnetization and frequency dependence in the peak of ac
susceptibility. The presence of a spin or cluster -glass like state is ruled
out by the absence of field divergence in third harmonic of ac susceptibility
and zero field cooled memory. This indicates that the observed spin-glass like
features are possibility 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 $\tau_0\approx 10^{-9}$ s. This together with
the existence of an AT line in H-T space indicates the existence of significant
inter-cluster interaction among these ferromagnetic-clusters.
