In this paper, we study a parabolic free boundary problem in an exterior domain

Here, a belongs to the interval (-1,0), K is a (given) convex compact set in Rⁿ, Ω={u>0}⊃K×(0,∞) is an unknown set, and F denotes a fully nonlinear operator. Assuming a suitable condition on the initial value u₀, we prove the existence of a nonnegative quasiconcave solution to the aforementioned problem, which exhibits monotone non-decreasing behavior over time.