Opuscula Math. 32, no. 3 (2012), 487-504
http://dx.doi.org/10.7494/OpMath.2012.32.3.487
Opuscula Mathematica
Global well-posedness and scattering for the focusing nonlinear Schrödinger equation in the nonradial case
Abstract. The energy-critical, focusing nonlinear Schrödinger equation in the nonradial case reads as follows: \[i\partial_t u = -\Delta u -|u|^{\frac{4}{N-2}}u,\quad (x,0)=u_0 \in H^1 (\mathbb{R}^N),\quad N\geq 3.\] Under a suitable assumption on the maximal strong solution, using a compactness argument and a virial identity, we establish the global well-posedness and scattering in the nonradial case, which gives a positive answer to one open problem proposed by Kenig and Merle [Invent. Math. 166 (2006), 645–675].
Keywords: critical energy, focusing Schrödinger equation, global well-posedness, scattering.
Mathematics Subject Classification: 35Q40, 35Q55.
- Pigong Han
- Chinese Academy of Sciences, Academy of Mathematics and Systems Science, Beijing 100190, China
- Received: 2011-06-11.
- Revised: 2011-08-04.
- Accepted: 2011-08-04.
