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

References in corpus (1)

• A Microscopic Convexity Principle for Nonlinear Partial Differential Equations