\(L^p\) norms of nonnegative Schrödinger heat semigroup and the large time behavior of hot spots
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 \(t\).
scientific article; zbMATH DE number 6017628