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 $$\begin{cases} F(D^2u)-\partial_tu=u^aχ_{\{u>0\}}&\text{in }(\mathbb R^n\setminus K)\times(0,\infty),\\ u=u_0&\text{on }\{t=0\},\\ |\nabla u|=u=0&\text{on }\partialΩ\cap(\mathbb R^n\times(0,\infty)),\\ u=1&\text{in }K\times[0,\infty).\end{cases}$$ Here, $a$ belongs to the interval $(-1,0)$, $K$ is a (given) convex compact set in $\mathbb R^n$, $Ω=\{u>0\}\supset K\times(0,\infty)$ 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.

Theorems & Definitions (22)

• Definition 1
• Definition 2
• Remark 1
• Theorem 1
• Proposition 1
• Lemma 1
• proof
• Proposition 2
• proof : Proof of Proposition \ref{['prop:envelope-subsol']}
• Lemma 2