Parabolic power concavity and parabolic boundary value problems

Abstract

This paper is concerned with power concavity properties of the solution to the parabolic boundary value problem \begin{equation} \tag{$P$} \left\{\begin{array}{ll} \partial_t u=Δu +f(x,t,u,\nabla u) & \mbox{in}\quadΩ\times(0,\infty),\vspace{3pt}\\ u(x,t)=0 & \mbox{on}\quad\partial Ω\times(0,\infty),\vspace{3pt}\\ u(x,0)=0 & \mbox{in}\quadΩ, \end{array} \right. \end{equation} where $Ω$ is a bounded convex domain in ${\bf R}^n$ and $f$ is a nonnegative continuous function in $Ω\times(0,\infty)\times{\bf R}\times{\bf R}^n$. We give a sufficient condition for the solution of $(P)$ to be parabolically power concave in $\barΩ\times[0,\infty)$.