The dynamics of the NLS with the combined terms in five and higher dimensions

Abstract

In this paper, we continue the study in \cite{MiaoWZ:NLS:3d Combined} to show the scattering and blow-up result of the solution for the nonlinear Schrödinger equation with the energy below the threshold $m$ in the energy space $H^1({\mathbb{R}}^d)$, iu_t + Δu = -|u|^{4/(d-2)}u + |u|^{4/(d-1)}u, \; d\geq 5. \tag{CNLS} The threshold is given by the ground state $W$ for the energy-critical NLS: $i{u}_t+\Delta u=-|u{|}^{4/(d-2)}u$. Compared with the argument in \cite{MiaoWZ:NLS:3d Combined}, the new ingredient is that we use the double duhamel formula in \cite{Kiv:Clay Lecture, TaoVZ:NLS:mass compact} to lower the regularity of the critical element in $L_t^{\infty }{H}_x^1$ to $L^{\infty }{\dot{H}}_x^{-\epsilon }$ for some $\epsilon >0$ in five and higher dimensions and obtain the compactness of the critical element in $L_x^2$, which is used to control the spatial center function $x(t)$ of the critical element and furthermore used to defeat the critical element in the reductive argument.

To publish in: Some Topics in Harmonic Analysis and Applications, Pages265-298, Advanced Lectures in Mathematics, ALM34, Higher Education Press, Beijing, and International Press, USA, 2015

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Cited by in corpus (2)

• Global well-posedness and scattering for nonlinear Schrödinger equations with combined nonlinearities in the radial case
• Scattering for the mass super-critical perturbations of the mass critical nonlinear Schrödinger equations