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Global well-posedness of the Cauchy problem for nonlinear Schrödinger-type equations. (English) Zbl 1129.35074

The authors are concerned with the well-posedness of the Cauchy problem for the nonlinear Schrödinger-like equation \[ \begin{aligned} iu_{t}+ (-\Delta)^mu= f(u), &\quad x\in\mathbb{R}^{n},\;t\in\mathbb{R},\\ u(0)=\psi, &\quad x\in\mathbb{R}^{n},\end{aligned} \] where \(m\geq1\) is an integer and \(f\in C\left( \mathbb{C},\mathbb{C}\right) .\) Under suitable assumptions regarding the nonlinear term, it is proved that the above Cauchy problem generates a global flow in \(L^{2}\) and \(H^{m}.\) In order to establish global well-posedness results under minimal smoothness restrictions on the nonlinear term, a suitable selection of working spaces based on a scaling technique is made, and a series of a priori estimates in the framework of Besov spaces is derived by using Strichartz estimates. The techniques suggested in the paper can be also extended to study scattering problems for the above problem and other classes of higher order evolution equations.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35B65 Smoothness and regularity of solutions to PDEs
37L05 General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations
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