Parabolic power concavity and parabolic boundary value problems

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=\Delta u +f(x,t,u,\nabla u) & \mbox{in}\quad\Omega\times(0,\infty),\vspace{3pt}\ u(x,t)=0 & \mbox{on}\quad\partial \Omega\times(0,\infty),\vspace{3pt}\ u(x,0)=0 & \mbox{in}\quad\Omega, \end{array} \right. \end{equation} where $\mathrm{\Omega }$ is a bounded convex domain in ${\mathbf{R}}^n$ and $f$ is a nonnegative continuous function in $\mathrm{\Omega }\times (0,\mathrm{\infty })\times \mathbf{R}\times {\mathbf{R}}^n$. We give a sufficient condition for the solution of $(P)$ to be parabolically power concave in $\bar{\mathrm{\Omega }}\times [0,\mathrm{\infty })$.