The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher

Abstract

We consider the focusing energy-critical nonlinear Schrödinger equation $i{u}_t+\Delta u=-|u{|}^{\frac{4}{d-2}}u$ in dimensions $d\ge 5$. We prove that if a maximal-lifespan solution $u:I\times\R^d\to \C$ obeys ${\sup }_{t\in I}\Vert \nabla u(t){\Vert }_2<\Vert \nabla W{\Vert }_2$, then it is global and scatters both forward and backward in time. Here $W$ denotes the ground state, which is a stationary solution of the equation. In particular, if a solution has both energy and kinetic energy less than those of the ground state $W$ at some point in time, then the solution is global and scatters. We also show that any solution that blows up with bounded kinetic energy must concentrate at least the kinetic energy of the ground state. Similar results were obtained by Kenig and Merle in \cite{Evian, kenig-merle} for spherically symmetric initial data and dimensions $d=3,4,5$.

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Cited by in corpus (7)

• Global wellposedness and scattering for the focusing energy-critical nonlinear Schrodinger equations of fourth order in the radial case
• Energy-supercritical NLS: critical ${\dot{H}}^s$-bounds imply scattering
• The radial defocusing energy-supercritical NLS in dimension four
• The linear profile decomposition for the fourth order Schrödinger equation
• The linear profile decomposition for the Airy equation and the existence of maximizers for the Airy Strichartz inequality
• The nonlinear Schrödinger equations with combined nonlinearities of power-type and Hartree
• On the mass-critical generalized KdV equation