Scattering and blow-up criteria for 3D cubic focusing nonlinear inhomogeneous NLS with a potential

Abstract

In this paper, we consider the 3d cubic focusing inhomogeneous nonlinear Schrödinger equation with a potential $$i{u}_t+\Delta u-Vu+|x{|}^{-b}|u{|}^{2}u=0,(t,x)\in \mathbf{R}\times {\mathbf{R}}^3,$$ where $0<b<1$. We first establish global well-posedness and scattering for the radial initial data $u_0$ in $H^1({\mathbf{R}}^3)$ satisfying $M(u_0{)}^{1-s_c}E(u_0{)}^{s_c}<\mathcal{E}$ and $\Vert u_0{\Vert }_{L^2}^{2(1-s_c)}\Vert H^{\frac{1}{2}}{u}_0{\Vert }_{L^2}^{2s_c}<\mathcal{K}$ provided that $V$ is repulsive, where $\mathcal{E}$ and $\mathcal{K}$ are the mass-energy and mass-kinetic of the ground states, respectively. Our result extends the results of Hong \cite{H} and Farah-Guzm$\acute{\mathrm{a}}$n \cite{FG1} with $b\in (0,\frac{1}{2})$ to the case $0<b<1$. We then obtain a blow-up result for initial data $u_0$ in $H^1({\mathbf{R}}^3)$ satisfying $M(u_0{)}^{1-s_c}E(u_0{)}^{s_c}<\mathcal{E}$ and $\Vert u_0{\Vert }_{L^2}^{2(1-s_c)}\Vert H^{\frac{1}{2}}{u}_0{\Vert }_{L^2}^{2s_c}>\mathcal{K}$ if $V$ satisfies some additional assumptions.0}\|_{L^{2}}^{2(1-s_{c})}\|H^{\frac{1}{2}}u_{0}\|_{L^{2}}^{2s_{c}}>\mathcal{K}$if$V$ satisfies some additional assumptions.

arXiv admin note: text overlap with arXiv:1810.07104

References in corpus (6)

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• Minimal mass blow-up solutions for the $L^2$ critical NLS with inverse-square potential
• Scattering theory in a weighted $L^2$ space for a class of the defocusing inhomogeneous nonlinear Schrödinger equation
• The focusing cubic NLS with inverse-square potential in three space dimensions
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• Blowup of $H^1$ solutions for a class of the focusing inhomogeneous nonlinear Schrödinger equation