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