  We seek for self-similar solutions describing the time-dependent evolution of
self-gravity systems with either spherical symmetry or axisymmetric disk
geometry. By assuming self-similar variable $x\equiv r/at$ where $a$ is
isothermal sound speed we find self-similar solutions extending from the
initial instant $t=0$ to the final stage $t\to \infty$ using standard
semi-analytical methods. Different types of solutions are constructed, which
describe overall expansion or collapse, envelope expansion with core collapse
(EECC), the formation of central rotationally supported quasi-equilibrium disk
as well as shocks. Though infinitely many, these self-similarity solutions have
similar asymptotic behaviors which may impose diagnosis on the velocity and
density structures in astrophysical systems.
