LpLp 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 potentialequation(P){∂tu=Δu−V(|x|)uin RN×(0,∞),u(x,0)=ϕ(x)in RN, where ∂t=∂/∂t∂t=∂/∂t, N⩾3N⩾3, ϕ∈L2(RN)ϕ∈L2(RN), and V=V(|x|)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+VH=−Δ+V is nonnegative on L2(RN)L2(RN), and give the exact power decay rates of LqLq norm (q⩾2)(q⩾2) of the solution e−tHϕe−tHϕ of (P) as t→∞t→∞. Furthermore we study the large time behavior of the solution of (P) and its hot spots.