Large time behavior of solutions of the heat equation with inverse square potential

Kazuhiro Ishige; Asato Mukai

doi:10.3934/dcds.2018176

Article Contents

2018, Volume 38, Issue 8: 4041-4069. Doi: 10.3934/dcds.2018176

This issue Previous Article On fractional Hardy inequalities in convex sets Next Article The regularity of solutions to some variational problems, including the p-Laplace equation for 3≤p < 4

Large time behavior of solutions of the heat equation with inverse square potential

Kazuhiro Ishige^, and
Asato Mukai

Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan

* Corresponding author: Kazuhiro Ishige
* Corresponding author: Kazuhiro Ishige

Received: November 2017

Revised: March 2018

Published: May 2018

The first author is partially supported by the Grant-in-Aid for Scientific Research (A)(No. 15H02058) from Japan Society for the Promotion of Science

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Abstract

Let $L: = -Δ+V$ be a nonnegative Schrödinger operator on $L^2({\bf R}^N)$, where $N≥ 2$ and $V$ is a radially symmetric inverse square potential. In this paper we assume either $L$ is subcritical or null-critical and we establish a method for obtaining the precise description of the large time behavior of $e^{-tL}\varphi$, where $\varphi∈ L^2({\bf R}^N, e^{|x|^2/4}\, dx)$.

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Mathematics Subject Classification: Primary: 35K15, 35B40; Secondary: 35K67.

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