# Energy-supercritical NLS: critical $\dot H^s$-bounds imply scattering

Mathematics – Analysis of PDEs

Scientific paper

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## Details Energy-supercritical NLS: critical $\dot H^s$-bounds imply scattering Energy-supercritical NLS: critical $\dot H^s$-bounds imply scattering

Scientific paper

We consider two classes of defocusing energy-supercritical nonlinear
Schr\"odinger equations in dimensions $d\geq 5$. We prove that if the solution
$u$ is apriorily bounded in the critical Sobolev space, that is, $u\in L_t^\infty \dot H^{s_c}_x$, then $u$ is global and scatters.

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