Mathematics > Analysis of PDEs
[Submitted on 16 Oct 2018]
Title:Scattering and blowup for L^{2}-supercritical and \dot{H}^{2}-subcritical biharmonic NLS with potentials
View PDFAbstract:We mainly consider the focusing biharmonic Schrödinger 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}}.
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