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

$$\begin{aligned} (P)\quad \left\{ \begin{array}{l@{\quad }l} \partial _t u=\varDelta u +f(x,t,u,\nabla u) &{} \text{ in }\quad \varOmega \times (0,\infty ),\\ u(x,t)=0 &{} \text{ on }\quad \partial \varOmega \times (0,\infty ),\\ u(x,0)=0 &{} \text{ in }\quad \varOmega , \end{array} \right. \end{aligned}$$

where $\varOmega$ is a bounded convex domain in $\mathbf{R}^n$ and $f$ is a nonnegative continuous function in $\varOmega \times (0,\infty )\times \mathbf{R}\times \mathbf{R}^n$. We give a sufficient condition for the solution of $(P)$ to be parabolically power concave in $\overline{\varOmega }\times [0,\infty )$.