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Non-preservation of $α$-concavity for the porous medium equation in higher dimensions

Xi Sisi Shen, Pranay Talla

Abstract

In this short note, we prove that $α$-concavity of the pressure is not preserved for the porous medium equation in dimensions $n=3$ and higher for any $α\in [0,1]\backslash \{\frac{1}{2}\}$. Together with the result of Chau-Weinkove for $n=2$, this fully resolves an open problem posed by Vásquez on whether pressure concavity is preserved in general for the porous medium equation.

Non-preservation of $α$-concavity for the porous medium equation in higher dimensions

Abstract

In this short note, we prove that -concavity of the pressure is not preserved for the porous medium equation in dimensions and higher for any . Together with the result of Chau-Weinkove for , this fully resolves an open problem posed by Vásquez on whether pressure concavity is preserved in general for the porous medium equation.
Paper Structure (2 sections, 2 theorems, 37 equations)

This paper contains 2 sections, 2 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Non-preservation of interior concavity

Key Result

Theorem 1

Let B be the open unit ball in $\mathbb{R}^n$ centered at the origin. For all $n$, given $\alpha \in [0,1] \backslash \{\frac{1}{2}\}$, there exists $v_0\in C^{\infty}(\overline{B})$ which is strictly positive on B and vanished on $\partial B$ with the following properties:

Theorems & Definitions (4)

  • Theorem 1
  • Lemma 1
  • proof
  • proof : Proof of Theorem \ref{['main_thm']}