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Global well-posedness, scattering and blow-up for the energy-critical, focusing, nonlinear Schrödinger equation in the radial case. (English) Zbl 1115.35125

Initial value problem for nonlinear Schrödinger equation \[ i\partial _t u + \Delta u\pm | u | ^{\frac{4}{N - 2}}u = 0, \quad (x,t) \in \mathbb R^N\times \mathbb R, \]
\[ u| _{t = 0} = u_0 \in \dot H^1(\mathbb R^N), \] is considered where the \(-\) sign corresponds to the defocusing problem, while the \(+\) sign corresponds to the focusing problem. The authors show: if \(E(u_0 ) < E(W),\| {u_0 } \| _{\dot H^1}< \| W \| _{\dot H^1}\), \(N = 3,4,5\) and \(u_0 \) is radial, then the solution \(u\) with data \(u_0 \) at \(t = 0\) is defined for all time and there exists \(u_{0, + }\), \(u_{0, - } \) in \(\dot H^1\) such that \[ \lim _{t \to + \infty} \| u( t) - e^{it\Delta}u_{0, + }\| _{\dot H^1}= 0, \quad \lim_{t \to - \infty} \| u( t) - e^{it\Delta}u_{0, - }\| _{\dot H^1}= 0. \] Here \(W(x) = 1 / (1 + | x | ^2 / N(N - 2))^{\frac{N - 2}{2}}\) is in \(\dot H^1(\mathbb R^N)\) and solves the elliptic equation \[ \Delta W + | W| ^{\frac{4}{N - 2}}W = 0. \] The authors also show that for \(u_0 \) radial, \(| x| u_0 \in L^2(\mathbb R^N)\), \(E(u_0 ) < E(W)\), but \(\| {u_0 }\| _{\dot H^1}>\| W\| _{\dot H^1}\), the solution must break down in finite time.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35B40 Asymptotic behavior of solutions to PDEs
35P25 Scattering theory for PDEs
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