Parabolic power concavity and parabolic boundary value problems

Authors
Type
Preprint
Publication Date
Jul 24, 2013
Submission Date
Jul 24, 2013
Identifiers
arXiv ID: 1307.6482
Source
arXiv
License
Yellow
External links

Abstract

This paper is concerned with power concavity properties of the solution to the parabolic boundary value problem $$\tag{P} \left\{\begin{array}{ll} \partial_t u=\Delta u +f(x,t,u,\nabla u) & \mbox{in}\quad\Omega\times(0,\infty),\vspace{3pt}\\ u(x,t)=0 & \mbox{on}\quad\partial \Omega\times(0,\infty),\vspace{3pt}\\ u(x,0)=0 & \mbox{in}\quad\Omega, \end{array} \right.$$ 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 $\bar{\Omega}\times[0,\infty)$.

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