Abstract
We prove scattering of solutions of the loglog energy-supercritical Schrödinger equation $i \partial _{t} u + \triangle u = |u|^{\frac{4}{n-2}} u g(|u|)$ with $g(|u|) := \log ^{\gamma } {( \log{(10+|u|^{2})} )}$, $0 < \gamma < \gamma _{n}$, n ∈ {3, 4, 5}, and with radial data $u(0) := u_{0} \in \tilde{H}^{k}:= \dot{H}^{k} (\mathbb{R}^{n})\,\cap\,\dot{H}^{1} (\mathbb{R}^{n})$, where $\frac{n}{2} \geq k> 1 \left(\text{resp.}\,\frac{4}{3}> k > 1\right)$ if n ∈ {3, 4} (resp. n = 5). The proof uses concentration techniques (see e.g., [ 2, 12]) to prove a long-time Strichartz-type estimate on an arbitrarily long time interval J depending on an a priori bound of some norms of the solution, combined with an induction on time of the Strichartz estimates in order to bound these norms a posteriori (see e.g., [ 8, 10]). We also revisit the scattering theory of solutions with radial data in $\tilde{H}^{k}$, $k> \frac{n}{2}$, and n ∈ {3, 4}; more precisely, we prove scattering for a larger range of $\gamma$ s than in [ 10]. In order to control the barely supercritical nonlinearity for nonsmooth solutions, that is, solutions with data in $\tilde{H}^{k}$, $k \leq \frac{n}{2}$, we prove some Jensen-type inequalities.
Long-Time Asymptotics for the Integrable Nonlocal Focusing Nonlinear Schrödinger Equation for a Family of Step-Like Initial Data