## 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 ^{-tH}φ of (P) as t→∞. Furthermore we study the large time behavior of the solution of (P) and its hot spots.

Original language | English |
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Pages (from-to) | 2695-2733 |

Number of pages | 39 |

Journal | Journal of Functional Analysis |

Volume | 262 |

Issue number | 6 |

DOIs | |

Publication status | Published - 2012 Mar 15 |

## Keywords

- Hot spots
- L -L estimates
- Schrödinger semigroups

## ASJC Scopus subject areas

- Analysis

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