@phdthesis{oai:sucra.repo.nii.ac.jp:00019597,
 author = {浜野, 大},
 month = {},
 note = {106p, In this paper, we deal with nonlinear Schrödinger system (NLS) in the mass-subcritical case and nonlinear Schrödinger equation with a potential (NLSV ) (or (NLSγ)) in the inter-critical case. We consider time behavior of solutions to these equations. For (NLS), we define a scattering threshold, by focusing structure of the nonlinearity, which corresponds to the best constant of small data scattering. We investigate a property of a solution on the threshold and an optimizing sequence of the threshold. For (NLSV ), we prove a scattering result, a blow-up or grow-up result, and a blow-up result below the ground state without a potential. Then, we show existence of a “radial” ground state and characterize the “radial” ground state by the virial functional. By using the “radial” ground state, we get a global well-posedness of (NLSV ). For (NLSγ), we show blow-up results. Moreover, we obtain equivalence of conditions on initial data below the ground state without a potential by utilizing the global well-posedness results and the blow-up result., 1. Introduction 2
1.1. Nonlinear Schrödinger equation 2
1.2. Nonlinear Schrödinger system 11
1.3. Nonlinear Schrödinger equation with a potential 19
1.4. Organization of the paper 27
2. Preliminaries 27
2.1. Notations 28
2.2. Some tools 29
3. Proof of theorems for NLS system 30
3.1. Notations for Section 3 30
3.2. Some tools for Section 3 31
3.3. Local well-posedness 34
3.4. Nonpositive energy implies failure of scattering 38
3.5. Stability 39
3.6. Properties of Lv0 and ℓtv0 42
3.7. Linear profile decomposition 45
3.8. Control of vanishing 50
3.9. Proof of Main theorems 1.39, 1.41, and 1.42 54
3.10. Study of related optimization problems 65
3.11. Proof of corollaries of Theorem 1.44 68
4. Proof of theorems for NLS with a potential 69
4.1. Some tools for Section 4 69
4.2. Proof of Main theorem 1.56 70
4.3. Proof of Main theorem 1.60 86
4.4. Proof of Theorem 1.62 90
4.5. Proof of Main theorem 1.64 90
4.6. Proof of Theorem 1.67 96
4.7. Proof of Main theorem 1.74 97
Acknowledgements 101
Reference 101, 指導教員 : 町原秀二, text, application/pdf},
 school = {埼玉大学},
 title = {Time behavior of solutions to nonlinear Schrödinger equations},
 year = {2021},
 yomi = {ハマノ, マサル}
}