\(L^p\) norms of nonnegative Schrödinger heat semigroup and the large time behavior of hot spots
type: Item
instance of: scholarly article
title: \(L^p\) norms of nonnegative Schrödinger heat semigroup and the large time behavior of hot spots
zbMATH Open document ID: 1252.35066
author: Kazuhiro Ishige
author: Yoshitsugu Kabeya
published in: Journal of Functional Analysis
publication date: 2012-03-22T00:00:00Z
review text: The study of the large time behavior of solutions to the Cauchy problem for a heat equation with a potential is a never ending story. The Cauchy problem of interest in \(\mathbb{R}^n\times (0,\infty)\) is \[ \partial_t u-\Delta u+ V(|x|)u= 0,\quad u(0,x)= \phi(x), \] where the radially symmetric potential \(V= V(|x|)\) is smooth and non-positive. Its asymptotic behavior is \(V(|x|)= \omega|x|^{-2}(1+ o(1))\) for \(|x|\to \infty\). Here, \(\omega\in(-\omega^*,0]\) and \(\omega^*= (n-2)^2/4\). The operator \(H:= -\Delta+ V\) is supposed to be a nonnegative Schrödinger operator on \(L^2(\mathbb{R}^n)\). The behavior of the solution depends heavily on the constant \(\omega\) in the potential \(V\). The authors are interested in the following issues the decay rate of the \(L^q\)-norm of \(u(t,\cdot)\) for \(t\to\infty\) with \(q\geq 2\) and the large time behavior of the solution and its hot spots. The authors recall for \(H\) the classification into subcritical, strongly subcritical, critical and supercritical Schrödinger operators. The main results of the paper can be summarized as follows. 1. Upper and lower estimates for the norm \(\|\exp(-tH)\|_{q,2}\) of the solution operator \(\exp(-tH)\) as mapping from \(L^2\) into \(L^q\) are given. 2. The long time behavior of the norm \(\| u(t,\cdot)\|_{L^2}\) of solutions to the above Cauchy problem is described for subcritical Schrödinger operators if the datum \(\phi\) belongs to \(L^2(\mathbb{R}^n,\exp(|x|^2/4)\,dx)\). 3. The hot spots are described for solutions to the above Cauchy problem with subcritical Schrödinger operators. Under suitable assumptions the authors prove, that the hot spots are curves for large \(t\).
reviewed by: Michael Reissig
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35K15
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 35J10
zbMATH DE Number: 6017628
zbMATH Keywords: \(L^p\)-\(L^q\) estimates
zbMATH Keywords: supercritical Schrödinger operators
MaRDI profile type: Publication
full work available at URL: j.jfa.2011.12.024
OpenAlex ID: W2048216109
cites work: Movement of hot spots in Riemannian manifolds
cites work: \(L^ p\) norms of non-critical Schrödinger semigroups
cites work: Movement of hot spots on the exterior domain of a ball under the Neumann boundary condition
cites work: Movement of hot spots on the exterior domain of a ball under the Dirichlet boundary condition
cites work: Decay rates of the derivatives of the solutions of the heat equations in the exterior domain of a ball
cites work: Hot spots for the heat equation with a rapidly decaying negative potential
cites work: Hot spots for the two dimensional heat equation with a rapidly decaying negative potential
cites work: Movement of hot spots over unbounded domains in \({\mathbb{R}}^ N\)
cites work: Eigenvalue problems in the whole space with radially symmetric weight
cites work: On bounded entire solutions of semilinear elliptic equations
cites work: Critical exponent for the bipolar blowup in a semilinear parabolic equation
cites work: Positive solutions and large time behaviors of Schrödinger semigroups, Simon's problem
cites work: Large time asymptotics for fundamental solutions of diffusion equations
cites work: Structure of positive solutions to \((-\Delta +V)u=0\) in \(R^ n\)
cites work: Radial entire solutions of the linear equation \(\Delta u+\lambda p(| x|)u=0\)
cites work: On positive solutions of second-order elliptic equations, stability results, and classification
cites work: Criticality and ground states for second-order elliptic equations
cites work: On criticality and ground states of second order elliptic equations. II
cites work: Large time behavior of the \(L^ p\) norm of Schrödinger semigroups
cites work: Global bounds of Schrödinger heat kernels with negative potentials
cites work: Subcriticality, positivity, and gaugeability of the Schrödinger operator
cites work: Subcriticality and Gaugeability of the Schrodinger Operator
DOI: 10.1016/J.JFA.2011.12.024
description: scientific article; zbMATH DE number 6017628