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

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

2011-12-20

arxiv.org/abs/1112.4618v1

Mathematics

Analysis of PDEs

33pages, 1 figure

Scientific paper

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 Schr\"{o}dinger equation with the energy below the threshold $m$ in the energy space $H^1(\R^d)$, iu_t + \Delta u = -|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: $iu_t + \Delta u = -|u|^{4/(d-2)}u$. 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 $L^{\infty}_tH^1_x$ to $L^{\infty}\dot H^{-\epsilon}_x$ for some $\epsilon>0$ in five and higher dimensions and obtain the compactness of the critical element in $L^2_x$, 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.

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