L p norms of nonnegative Schrödinger heat semigroup and the large time behavior of hot spots

Kazuhiro Ishige, Yoshitsugu Kabeya

Research output: Contribution to journalArticlepeer-review

14 Citations (Scopus)

Abstract

This paper is concerned with the Cauchy problem for the heat equation with a potential(P){∂tu=δu-V(|x|)uin R N×(0,∞),u(x,0)=φ(x)in R N, where ∂ t=∂/∂t, N≥3, φ∈L 2(R N), and V=V(|x|) is a smooth, nonpositive, and radially symmetric function having quadratic decay at the space infinity. In this paper we assume that the Schrödinger operator H=-δ+V is nonnegative on L 2(R N), and give the exact power decay rates of L q norm (q≥2) of the solution e -tHφ of (P) as t→∞. Furthermore we study the large time behavior of the solution of (P) and its hot spots.

Original languageEnglish
Pages (from-to)2695-2733
Number of pages39
JournalJournal of Functional Analysis
Volume262
Issue number6
DOIs
Publication statusPublished - 2012 Mar 15

Keywords

  • Hot spots
  • L -L estimates
  • Schrödinger semigroups

ASJC Scopus subject areas

  • Analysis

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