| Notes on Global Existence for the Nonlinear Schrdinger Equation Involves Derivative |
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| 摘 要： | In this paper, we consider the scattering for the nonlinear Schr¨odinger equation with small,smooth, and localized data. In particular, we prove that the solution of the quadratic nonlinear Schr¨odinger equation with nonlinear term |u|2involving some derivatives in two dimension exists globally and scatters. It is worth to note that there exist blow-up solutions of these equations without derivatives. Moreover, for radial data, we prove that for the equation with p-order nonlinearity with derivatives, the similar results hold for p ≥2d+32d-1and d ≥ 2, which is lower than the Strauss exponents.
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| 关 键 词： | 非线性薛定谔方程 衍生产品 整体存在性 票据 二次非线性 非线性项 衍生品 散射 |
Notes on global existence for the nonlinear Schrödinger equation involves derivative |
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| Authors: | Shi Yang Zheng |
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| Institution: | 1. School of Mathematics and Systems Science, Beihang University, Beijing, 100191, P. R. China
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| Abstract: | In this paper, we consider the scattering for the nonlinear Schrödinger equation with small, smooth, and localized data. In particular, we prove that the solution of the quadratic nonlinear Schrödinger equation with nonlinear term |u|2 involving some derivatives in two dimension exists globally and scatters. It is worth to note that there exist blow-up solutions of these equations without derivatives. Moreover, for radial data, we prove that for the equation with p-order nonlinearity with derivatives, the similar results hold for \(p \geqslant \tfrac{{2d + 3}} {{2d - 1}} \) and d ≥ 2, which is lower than the Strauss exponents. |
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| Keywords: | SchrSdinger equation global well-posedness scatter small data quadratic |
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