Layzer-Irvine equation [38] when the interaction is neglected. In the homogeneous case, the DE component flows progressive out of the overdensity which leads the energy conservation law invalid in the collapsing system [43]. It is expected in the models of DE interacting with DM, if the DE is distributed inhomogeneously, it may cluster along with the DM and thus we can avoid the energy non-conservation problem. To see whether this speculation is correct or not, we have further explored the dynamics of the DE that presents inhomogeneities at cluster scales and found that in the presence of the coupling between DE and DM, the time and dynamics required by the DE in the system to reach equilibrium is different from that of DM. This shows that in the collapsing system, DE does not fully cluster along with DM, even in the presence of the interaction between them. The energy is not strictly conserved in the collapsing system. This result is based on the linear perturbation formalism, where the DE perturbation can be neglected in the late universe [19]. 

We also developed a novel treatment of the spherical collapse model which consists of multi-fluids with different collapsing velocities. When DE does not trace DM in the spherical collapse, we studied the dependence of the cluster number counts on the coupling between DE and DM and on the DE inhomogeneities by using the Press-Schechter formalism. In most of our work we have assumed that the models are normalized to have practically the same σ$_{8}$ = 0 . 8 as given by present day observations. We have shown that when the coupling between dark sectors is proportional to the energy density of DE, there is a significant dependence of cluster number counts on the interaction. If the energy decays from DE to DM, the number of cluster counts increases. The dependence of cluster number counts on the DE inhomogeneities is negligible compared with the interaction influence. This in fact shows that the fluctuation of the DE field is small and the inhomogeneous DE plays little role in the virialization of the structure. For the coupling between DE and DM proportional to the energy density of DM and total dark sectors, we have observed that more energy decay from DE to DM leads more abundance of the cluster number counts compared with the ΛCDM model. In addition, the signature of the DE inhomogeneities is enhanced. This enhancement can be understood from the dynamics in which it shows that the DM gravitational effect pulls the inhomogeneous DE to move and collapse along with DM. 

We have further studied these models enforcing the normalization to reproduce the same abundance of galaxy clusters at redshift zero. The qualitative structure changes in choosing different normalizations [27]. Comparing with the interaction effects in fixing σ$_{8}$ , we have verified that the departures from the ΛCDM model are suppressed, the coupling between dark sectors while fixing the local abundance usually does not show up in the galaxy number counts, except when the interaction is proportional to the energy density of DE in the super mass bin. In fixing the local abundance, the influence caused by the inhomogeneous DE is enhanced with large deviations from the cosmological 

