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Global existence of small solutions to nonlinear Schrödinger equations. (English) Zbl 0886.35141

This paper is concerned with the global existence of small solutions to \[ i\partial _t u + \frac{1}{2}\Delta u = F(u,\nabla u,\bar{u},\nabla{\bar{u}}),\quad (t,x) \in \mathbb{R} \times \mathbb{R}^n,\;u(0,x) = \varepsilon_0 \phi(x), \;x \in \mathbb{R}^n, \] where \(n \geq 3\), \(\varepsilon_0\) is sufficiently small, \[ F=F(u,w,\bar{u},\bar{w})=\sum_{l_0 \leq |\alpha|+|\beta|+|\gamma|\leq l_1} \lambda_{\alpha,\beta,\gamma} u^{\alpha_1}\bar{u}^{\alpha_2}w^{\beta_j}\bar{w}^{\gamma_k}+ \sum_{|\beta|+|\gamma|=2}\lambda_{\beta,\gamma} w^{\beta_j}\bar{w}^{\gamma_k} \] with \(w=(w_j),1 \leq j \leq n,\lambda_{\alpha,\beta,\gamma},\lambda_{\beta,\gamma} \in C\) (complex numbers), \(l_0,l_1 \in N\) (integers), \(l_0=3,4\) for \(l_0=2\) for \(n \geq 5.\) The authors prove a unique existence of global solutions for \(\varepsilon_0\) sufficiently small and for \(n \geq 3.\) It is remarked that the gradient of \(F\) with respect to \(w\) is not assumed to be pure imaginary.
Reviewer: A.Tsutsumi (Osaka)

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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