Parabolic power concavity and parabolic boundary value problems

TL;DR

This paper investigates conditions under which solutions to certain parabolic boundary value problems exhibit power concavity, extending understanding of solution shape properties in convex domains with nonlinear source terms.

Contribution

It provides a sufficient condition ensuring the solution to a class of parabolic PDEs is parabolically power concave in convex domains.

Findings

Identifies a sufficient condition for power concavity of solutions.

Extends concavity results to nonlinear parabolic boundary value problems.

Applicable to problems with nonnegative continuous source functions.

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