  In this paper, we study a parabolic free boundary problem in an exterior
domain $$\begin{cases} F(D^2u)-\partial_tu=u^a\chi_{\{u>0\}}&\text{in }(\mathbb
R^n\setminus K)\times(0,\infty),\\ u=u_0&\text{on }\{t=0\},\\ |\nabla
u|=u=0&\text{on }\partial\Omega\cap(\mathbb R^n\times(0,\infty)),\\
u=1&\text{in }K\times[0,\infty).\end{cases}$$ 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.
