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- W2079291604 abstract "We consider the focusing energy-critical nonlinear Schrodinger equation $iu_t+Delta u = - |u|^{frac4{d-2}}u$ in dimensions $dgeq 5$. We prove that if a maximal-lifespan solution $u:ItimesR^dto C$ obeys $sup_{tin I}|nabla u(t)|_2<|nabla W|_2$, 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 $d=3,4,5$." @default.
- W2079291604 created "2016-06-24" @default.
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- W2079291604 date "2008-04-07" @default.
- W2079291604 modified "2023-10-18" @default.
- W2079291604 title "The focusing energy-critical nonlinear Schr" @default.
- W2079291604 hasPublicationYear "2008" @default.
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