The Cauchy problem of a focusing energy-critical nonlinear Schrödinger equation |
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| Institution: | 1. Department of Mathematics, Northwest Normal University, Lanzhou, 730070, PR China;2. School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, PR China;1. College of Mathematics and Software Science, Sichuan Normal University, Chengdu, 610066, China;2. College of Mathematics and Information, China west Normal University, Nanchong, 637002, China;1. School of Mathematics and Physics, University of Science and Technology Beijing, 30 Xueyuan Road, Haidian District, Beijing, 100083, PR China;2. Department of Mathematics, Henan Normal University, Xinxiang, 453007, PR China;1. Department of Mathematics, Northwest Normal University, Lanzhou 730070, China;2. School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China |
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| Abstract: | In this paper, we combine variational methods and harmonic analysis to discuss the Cauchy problem of a focusing nonlinear Schrödinger equation. We study the global well-posedness, finite time blowup and asymptotic behavior of this problem. By Hamiltonian property, we establish two types of invariant evolution flows. Then from one flow and the stability of classical energy-critical nonlinear Schrödinger equation, we find that the solution exists globally and scattering occurs. Finally, we get a precise blowup criterion of this problem for positive energy initial data via the other flow. |
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