Abstract
We consider the Cauchy problem for the one dimensional nonlinear dissipative Schrödinger equation with a cubic nonlinearity \(\lambda |u|^2u\), where \(\lambda \in {\mathbb {C}}\) with Im \(\lambda < 0\). We show that a relation between \(L^2\)-decay rate for the solution and a smoothness of the initial data. Our result improves the recent work of Hayashi–Li–Naumkin (Adv Math Phys Art. ID 3702738, 7, 2016) for the decay rate of \(L^2\).
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Ogawa, T., Sato, T. \(\mathbf{L^2}\)-decay rate for the critical nonlinear Schrödinger equation with a small smooth data. Nonlinear Differ. Equ. Appl. 27, 18 (2020). https://doi.org/10.1007/s00030-020-0621-3
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DOI: https://doi.org/10.1007/s00030-020-0621-3