Dissipative nonlinear Schrödinger equations for large data in one space dimension (Q2286204)

The author considers the Cauchy problem for the nonlinear Schrödinger equations \[ \begin{cases} i\frac{\partial u}{\partial t} +\frac{1}{2} \frac{\partial^2 u}{\partial x^2} = \lambda|u|^{p-1}u \\ u(0)=\phi \end{cases}, \] where \(1< p \leq 3, \lambda\in {\mathbb C}\), \(\mathrm{Im}(\lambda) < 0, \quad \frac{p-1}{2\sqrt{p}} |\Re(\lambda)|\leq |\mathrm{Im}(\lambda)|\). Let \(\mathcal{F}\) be the Fourier transform and \(H^s\) be the Sobolev space. For \(\phi \in\mathcal{F} H^\gamma, 1/2 < \gamma \leq 1\) global existence of the solution to the above Schrödinger equation is proved (without any smallness condition and regularity assumption for the data \(\phi\)). Under additional assumption \(p>\frac{1}{4} \left(7-2\gamma + \sqrt{(7-2\gamma)^2+8(2\gamma-1)} \right)\) asymptotic behavior for solutions is also obtained.