L ~p norms of nonnegative Schr?dinger heat semigroup and the large time behavior of hot spots

摘要

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 Schr?dinger 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.