We study a free boundary problem of parabolic equations with a pos-itive parameter τ included in the coefficient of the derivative with respect to the timevariable t. This problem arises from some reaction-diffusion system. We prove that, ifτ is large enough, the solution exists for 0 ＜ t ＜ +∞; while, if τ is small enough, thesolution exists only in finite time.