We present calculations that account for the infall of matter onto the heart of a starburst, a condition thought indispensable for causing the large accumulation of matter and the large detected rates of star formation. We center our attention on the rapid evolution of the remnant generated by the collective action of strong stellar winds and supernovae and particularly on its rapid approach toward a steady state, or pressure-confined, solution. Two different stationary solutions may result depending on the relative thickness of the infalling disk compared with the distance along the galaxy plane at which the reverse, or inner, shock acquires its stationary location. In both cases the infalling disk matter experiences a giant splash after crossing the outer shock to be ejected into lower pressure zones at large distances away from the heart of the starburst. Also, in both cases the giant disk matter splash ends up confining the central wind into a biconical flow with a stationary, narrow (~ a few hundred pc) central waist. The observational implications of both stationary solutions are discussed.