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

Abstract

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

24 pages