Critical nonlinear Schrödinger equations in higher space dimensions. (English) Zbl 1408.35173

Summary: We study the critical nonlinear Schrödinger equations \[ i\partial _{t}u+\frac{1}{2}\Delta u = \lambda | u|^{{2}/{n}}u, (t,x) \in \mathbb{R}^{+}\times \mathbb{R}^{n}, \] in space dimensions \(n\geq 4\), where \(\lambda \in \mathbb{R}\). We prove the global in time existence of solutions to the Cauchy problem under the assumption that the absolute value of Fourier transform of the initial data is bounded below by a positive constant. Also we prove the two side sharp time decay estimates of solutions in the uniform norm.


35Q55 NLS equations (nonlinear Schrödinger equations)
35B40 Asymptotic behavior of solutions to PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
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