  In order to investigate the inhomogeneity of the superconducting (SC) state
in the overdoped high-$T_{\rm c}$ cuprates, we have measured the magnetic
susceptibility, $\chi$, of La$_{2-x}$Sr$_x$CuO$_4$ (LSCO) single crystals in
the overdoped regime in magnetic fields parallel to the c-axis up to 7 T on
warming after zero-field cooling. It has been found for $x$ = 0.198 and 0.219
that the temperature dependence of $\chi$ in 1 T shows a plateau, that is,
$\chi$ is almost independent of temperature in a moderate temperature-range in
the SC state. Moreover, a so-called second peak in the magnetization curve has
markedly appeared in these crystals. These results indicate an anomalous
enhancement of the vortex pinning and strongly suggest the occurrence of a
$microscopic$ phase separation into SC and normal-state regions in the
overdoped high-$T_{\rm c}$ cuprates.
