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

This paper is concerned with the Cauchy problem for the heat equation with a potential ∂tu = u −V |x| u in RN ×(0,∞), u(x, 0) = φ(x) in RN, (P) where ∂t = ∂/∂t , N 3, φ ∈ L2(RN), 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 =− + V is nonnegative on L2(RN), and give the exact power decay rates of Lq 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. © 2011 Elsevier Inc. All rights reserved.