Convexity for a parabolic fully nonlinear free boundary problem with singular term

In this paper, we study a parabolic free boundary problem in an exterior domain $$\{\begin{array}{ll}F(D^{2}u)-{\mathrm{\partial }}_{t}u=u^a{\chi }_{\{u>0\}}& \text{in }({\mathbb{R}}^n\setminus K)\times (0,\mathrm{\infty }),\\ u=u_0& \text{on }\{t=0\},\\ \mathrm{\mid }\mathrm{\nabla }u\mathrm{\mid }=u=0& \text{on }\mathrm{\partial }\mathrm{\Omega }\cap ({\mathbb{R}}^n\times (0,\mathrm{\infty })),\\ u=1& \text{in }K\times [0,\mathrm{\infty })\mathrm{.}\end{array}$$ Here, $a$ belongs to the interval $(-1,0)$, $K$ is a (given) convex compact set in ${\mathbb{R}}^n$, $\mathrm{\Omega }=\{u>0\}\supset K\times (0,\mathrm{\infty })$ is an unknown set, and $F$ denotes a fully nonlinear operator. Assuming a suitable condition on the initial value $u_0$, we prove the existence of a nonnegative quasiconcave solution to the aforementioned problem, which exhibits monotone non-decreasing behavior over time.