Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4591326 | Journal of Functional Analysis | 2012 | 39 Pages |
Abstract
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.
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Kazuhiro Ishige, Yoshitsugu Kabeya,