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Zbl 1060.35131
Colliander, James E.; Keel, Markus; Staffilani, Gigliola; Takaoka, Hideo; Tao, Terence C.
Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $\Bbb R^3$.
(English)
[J] Commun. Pure Appl. Math. 57, No. 8, 987-1014 (2004). ISSN 0010-3640

The authors study the following initial value problem for a cubic defocusing nonlinear Schrödinger equation $$\gather i\partial_t\varphi(x,t)+ \Delta_x\varphi(x,t)= \vert\varphi(x, t)\vert^2 \varphi(x,t),\quad x\in\bbfR^3,\ t\ge 0,\\ \varphi(x,0)= \varphi_0(x)\in H^s(\bbfR^3).\endgather$$ Here $H^s(\bbfR^3)$ denotes the usual inhomogeneous Sobolev space. The authors prove global existence in $H^s(\bbfR^3)$ for $s> 4/5$.
[Messoud A. Efendiev (Berlin)]
MSC 2000:
*35Q55 NLS-like (nonlinear Schroedinger) equations
35Q05 Euler-Poisson-Darboux equation and generalizations
35B30 Dependence of solutions of PDE on initial and boundary data

Keywords: nonlinear Schrödinger equation; global existence; Cauchy problem

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