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 {F(D(2)u) - partial derivative(t)u = u(chi{u>0})(a) in (R-n\K) x (0,infinity), u = u(0 )on {t=0}, |del u| = u = 0 on partial derivative Omega boolean AND (R(n )x (0,infinity)), u = 1 in K x [0,infinity). Here, a belongs to the interval (-1,0), K is a (given) convex compact set in R-n, Omega={u > 0} superset of K x (0,infinity) 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.

In this paper, we study a parabolic free boundary problem in an exterior domain (Formula presented.) Here, a belongs to the interval (-1,0), K is a (given) convex compact set in Rn, Ω={u>0}⊃K×(0,∞) is an unknown set, and F denotes a fully nonlinear operator. Assuming a suitable condition on the initial value u0, we prove the existence of a nonnegative quasiconcave solution to the aforementioned problem, which exhibits monotone non-decreasing behavior over time.