We explore semi-complete self-similar solutions for the polytropic gas dynamics involving self-gravity under spherical symmetry, examine behaviours of the sonic critical curve, and present new asymptotic collapse solutions that describe `quasi-static asymptotic behaviours at small radii and large times. These new `quasi-static solutions with divergent mass density approaching the core can have self-similar oscillations. Earlier known solutions are summarized. Various semi-complete self-similar solutions involving such novel asymptotic solutions are constructed, either with or without a shock. In contexts of stellar core collapse and supernova explosion, a hydrodynamic model of a rebound shock initiated around the stellar degenerate core of a massive progenitor star is presented. With this dynamic model framework, we attempt to relate progenitor stars and the corresponding remnant compact stars: neutron stars, black holes, and white dwarfs.
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