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