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Luiz Gustavo Farah Diashttp://lattes.cnpq.br/8538404005712205Svetlana RoudenkoAdemir Pastor FerreiraAlex Javier Hernandez ArdilaJosé Felipe Linares RamirezPaulo Cupertino de Limahttp://lattes.cnpq.br/5533021450394783Luccas Cassimiro Campos2020-11-30T18:04:13Z2020-11-30T18:04:13Z2019-12-12http://hdl.handle.net/1843/34446https://orcid.org/0000-0001-5550-6082We show several results regarding long-time behavior of solutions to Schrödinger-type equations. For the focusing (classical) nonlinear Schrödinger (NLS) equation, we study solutions at the mass-energy threshold in the intercritical and energy-critical setting. We completely identify and classify the behavior of such solutions, showing that there is some rigidity in this regime. In the energy-critical setting, we extend the works of Duyckaerts and Merle [24] to dimensions N $\geq$ 6 (see also Li and Zhang [63] for a different approach), and in the intercritical range, we extend the work of Duyckaerts and Roudenko [25]. For the focusing inhomogeneous nonlinear Schrödinger (INLS) equation, we present a proof of scattering below the ground state, adapting the approach of Dodson and Murphy [22] to the INLS, and extending the previous results of Farah and Guzmán [31, 30]. We also discuss the behavior of solutions to the INLS that are above the mass-energy threshold. We give a dichotomy between scattering and blow-up in this scenario, and also some blow-up criteria. This chapter extends the works of Duyckaerts and Roudenko [26] to the INLS equation.Neste trabalho, apresentamos diversos resultados relacionados ao comportamento assintótico de soluções de equações do tipo Schrödinger. Para o caso clássico (e do tipo focusing) da equação de Schrödinger não-linear (NLS), descrevemos as soluções no limiar massa-energia, tanto no caso intercrítico quanto no caso H1-crítico. O comportamento dessas soluções é completamente classificado, mostrando que há uma certa rigidez quanto aos tipos de solução possíveis nesse regime. No contexto H1-crítico, estendemos o trabalho de Duyckaerts e Merle [24] para dimensões N $\geq$ 6 (c.f. Li e Zhang [63] para uma abordagem diferente), e no caso intercrítico, o trabalho de Duyckaerts e Roudenko [25]. Para a equação de Schrödinger não-linear e não-homogênea (INLS), apresentamos uma prova do scattering (espalhamento) abaixo do ground state (estado estacionário), adaptando a abordagem de Dodson e Murphy [22] para a INLS, bem como estendendo resultados anteriores de Farah e Guzmán [31, 30]. Discutimos também o comportamento de soluções da INLS que estão acima do limiar massa-energia. Exibimos um cenário em que há uma dicotomia entre scattering e blow-up (explosão), além de provar diferentes critérios de blow-up. Estendemos, assim, o trabalho de Duyckaerts e Roudenko [26] para a INLS.CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível SuperiorengUniversidade Federal de Minas GeraisPrograma de Pós-Graduação em MatemáticaUFMGBrasilICEX - INSTITUTO DE CIÊNCIAS EXATAShttp://creativecommons.org/licenses/by-nc-nd/3.0/pt/info:eu-repo/semantics/openAccessMatemática - TesesSchrodinger, Equação de - TesesEspalhamento - TesesExplosão (blow-up) - TesesNonlinear Schrödinger-type equationsGlobal behaviorScatteringBlow-upLong-time behavior of solutions to nonlinear Schrödinger-type equationsComportamento global de soluções de equações do tipo Schrödingerinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/doctoralThesisreponame:Repositório Institucional da UFMGinstname:Universidade Federal de Minas Gerais (UFMG)instacron:UFMGORIGINALPhD_Thesis_ficha.pdfPhD_Thesis_ficha.pdfapplication/pdf1698062https://repositorio.ufmg.br/bitstream/1843/34446/1/PhD_Thesis_ficha.pdf8906f75b8eee5e4a9c3a768d24c982deMD51CC-LICENSElicense_rdflicense_rdfapplication/rdf+xml; charset=utf-8811https://repositorio.ufmg.br/bitstream/1843/34446/2/license_rdfcfd6801dba008cb6adbd9838b81582abMD52LICENSElicense.txtlicense.txttext/plain; charset=utf-82119https://repositorio.ufmg.br/bitstream/1843/34446/3/license.txt34badce4be7e31e3adb4575ae96af679MD531843/344462020-11-30 15:04:13.161oai:repositorio.ufmg.br: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Repositório InstitucionalPUBhttps://repositorio.ufmg.br/oaiopendoar:2020-11-30T18:04:13Repositório Institucional da UFMG - Universidade Federal de Minas Gerais (UFMG)false
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