As a model for the Lymann alpha forest, the evolution of baryon clouds under the potential of cold dark matter, so-called minihalos, is examined by taking into account the time variation of diffuse UV flux. After minihalos settle into a stable equilibrium branch, they are gradually destabilized due to a decrease in UV flux; finally, they undergo a free-fall collapse, probably leading to the formation of low-mass galaxies. This behavior is related to the observed number density evolution of the Lyman alpha forest and the mass spectrum of minihalos. A new idea for the inverse effect in relation to this minihalo evolution is also presented.