Line-driven winds from hot stars and accretion disks are thought to adopt a unique, critical solution which corresponds to maximum mass loss rate and a particular velocity law. We show that in the presence of negative velocity gradients, radiative-acoustic (Abbott) waves can drive shallow wind solutions towards larger velocities and mass loss rates. Perturbations introduced downstream from the wind critical point lead to convergence towards the critical solution. By contrast, low-lying

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... ions cause evolution towards a mass-overloaded solution, developing a broad deceleration region in the wind. Such a wind differs fundamentally from the critical solution. For sufficiently deep-seated perturbations, overloaded solutions become time-dependent and develop shocks and shells.