The Hardy inequality and large time behaviour of the heat equation on $\mathbb{R}^{N-k}\times (0,\infty)^k$

TL;DR

This paper investigates the large time behavior of the heat equation with Hardy potential on corner spaces, establishing optimal decay rates and asymptotic expansions while extending previous results to more general geometries.

Contribution

It introduces an improved Hardy-Poincaré inequality, constructs a suitable functional framework, and extends decay rate results to higher-dimensional corner spaces without using spherical harmonics.

Findings

Optimal polynomial decay rates for solutions

First asymptotic term in $L^2$ norm identified

Decay rates improve with higher $k$ values

Abstract

In this paper we study the large time asymptotic behaviour of the heat equation with Hardy inverse-square potential on corner spaces $\mathbb{R}^{N-k}\times (0,\infty)^k$, $k\geq 0$. We first show a new improved Hardy-Poincar\'{e} inequality for the quantum harmonic oscillator with Hardy potential. In view of that, we construct the appropriate functional setting in order to pose the Cauchy problem. Then we obtain optimal polynomial large time decay rates and subsequently the first term in the asymptotic expansion of the solutions in $L^2(\mathbb{R}^{N-k}\times (0,\infty)^k)$. Particularly, we extend and improve the results obtained by V\'{a}zquez and Zuazua (J. Funct. Anal. 2000), which correspond to the case $k=0$, to any $k\geq 0$. We emphasize that the higher the value of $k$ the better time decay rates are. We employ a different and simplified approach than V\'{a}zquez and Zuazua, managing to remove the usage of spherical harmonics decomposition in our analysis.