Scattering and blowup for $L^{2}$-supercritical and $\dot{H}^{2}$-subcritical biharmonic NLS with potentials

TL;DR

This paper proves global well-posedness, scattering, and blow-up results for a radial focusing biharmonic Schrödinger equation with large potentials in high dimensions, extending understanding of supercritical and subcritical regimes.

Contribution

It establishes the first scattering results for the biharmonic NLS with large potentials in the supercritical/subcritical regime, including full Strichartz and smoothing estimates.

Findings

Proved global well-posedness and scattering for radial data under certain energy and mass conditions.

Established blow-up results for data exceeding the ground state thresholds.

Developed fundamental linear flow estimates with large potentials.

Abstract

We mainly consider the focusing biharmonic Schr\"odinger equation with a large radial repulsive potential $V(x)$: \begin{equation*} \left\{ \begin{aligned} iu_{t}+(\Delta^2+V)u-|u|^{p-1}u=0,\;\;(t,x) \in {{\bf{R}}\times{\bf{R}}^{N}}, u(0, x)=u_{0}(x)\in H^{2}({\bf{R}}^{N}), \end{aligned}\right. \end{equation*} If $N>8$, \ $1+\frac{8}{N}<p<1+\frac{8}{N-4}$ (i.e. the $L^{2}$-supercritical and $\dot{H}^{2}$-subcritical case ), and $\langle x\rangle^\beta \big(|V(x)|+|\nabla V(x)|\big)\in L^\infty$ for some $\beta>N+4$, then we firstly prove a global well-posedness and scattering result for the radial data $u_0\in H^2({\bf R}^N)$ which satisfies that $$ M(u_0)^{\frac{2-s_c}{s_c}}E(u_0)<M(Q)^{\frac{2-s_c}{s_c}}E_{0}(Q) \ \ {\rm{and}}\ \ \|u_{0}\|^{\frac{2-s_c}{s_c}}_{L^{2}}\|H^{\frac{1}{2}} u_{0}\|_{L^{2}}<\|Q\|^{\frac{2-s_c}{s_c}}_{L^{2}}\|\Delta Q\|_{L^{2}}, $$ where $s_c=\frac{N}{2}-\frac{4}{p-1}\in(0,2)$, $H=\Delta^2+V$ and $Q$ is the ground state of $\Delta^2Q+(2-s_c)Q-|Q|^{p-1}Q=0$. We crucially establish full Strichartz estimates and smoothing estimates of linear flow with a large poetential $V$, which are fundamental to our scattering results. Finally, based on the method introduced in \cite[T. Boulenger, E. Lenzmann, Blow up for biharmonic NLS, Ann. Sci. $\acute{E}$c. Norm. Sup$\acute{e}$r., 50(2017), 503-544]{B-Lenzmann}, we also prove a blow-up result for a class of potential $V$ and the radial data $u_0\in H^2({\bf R}^N)$ satisfying that $$ M(u_0)^{\frac{2-s_c}{s_c}}E(u_0)<M(Q)^{\frac{2-s_c}{s_c}}E_{0}(Q) \ \ {\rm{and}}\ \ \|u_{0}\|^{\frac{2-s_c}{s_c}}_{L^{2}}\|H^{\frac{1}{2}} u_{0}\|_{L^{2}}>\|Q\|^{\frac{2-s_c}{s_c}}_{L^{2}}\|\Delta Q\|_{L^{2}}. $$