Lp norms of nonnegative Schrodinger 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 {partial derivative(t)u = Delta u - V(vertical bar x vertical bar)u in R-N x (0, infinity), u(x, 0) = phi(x) in R-N, (P) where partial derivative(t) = partial derivative/partial derivative t, N >= 3, phi is an element of L-2(R-N), and V = V (vertical bar x vertical bar) is a smooth, nonpositive, and radially symmetric function having quadratic decay at the space infinity. In this paper we assume that the Schrodinger operator H = -Delta + 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) phi of (P) as t -> infinity. Furthermore we study the large time behavior of the solution of (P) and its hot spots. (C) 2011 Elsevier Inc. All rights reserved.