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Luiz Gustavo Farah Diashttp://lattes.cnpq.br/8538404005712205Carlos Manuel Guzmán JiménezHenrique de Melo Versieuxhttp://lattes.cnpq.br/3461776604785091Alan Bruno do Nascimento2023-11-20T11:57:27Z2023-11-20T11:57:27Z2020-02-28http://hdl.handle.net/1843/61123Neste trabalho estudamos a boa colocação local e global, bem como a explosão em tempo finito e espalhamento das soluções no espaço de energia do problema de valor inicial associado à equação de Schrödinger não-linear em R^3 i∂tu + Δu + |u|2u = 0. (1) No Capítulo 2 introduzimos ferramentas essenciais de análise funcional e análise harmônica, assim como propriedades da equação elíptica Δψ − ψ + ψ3 = 0,que são importantes para o entendimento dos fenômenos da equação de Schrödinger acima enunciados. No Capítulo 3 demonstramos a boa colocação local para a equação (1), por meio do Teorema do Ponto Fixo de Banach. No Capítulo 4, utilizamos o método de decomposição em perfis em H1(R3) para determinar a constante ótima para a desigualdade de Gagliardo-Nirenberg ∥f∥4L4 ≤ c ∥∇f∥3L2 ∥f∥L2 , um resultado valioso para o entendimento da dicotomia boa colocação global e explosão, demonstrada em seguida. Finalmente, no Capítulo 5, passamos ao estudo do comportamento assintótico das soluções globais de (1). Demonstramos que, abaixo do limiar massa-energia da equação elíptica associada, as soluções globais têm espalhamento.In this work we study the local and global well-posedness, as well as the finite time blow-up and scattering of the solutions in the energy space of the inicial value problem associated with the nonlinear Schrödinger equation in R^3 i∂tu + Δu + |u|2u = 0. (1) In Chapter 2, we introduce essential tools from functional and harmonic analysis, as well as properties of the elliptic equation Δψ − ψ + ψ3 = 0, which are important for understanding the phenomena of the Schrödinger equation above stated. In Chapter 3 we show the local well-posedness for the equation (1), via the Banach Fixed Point Theorem. In Chapter 4, we make use of the profile decomposition method in H^1(R^3) to determine the sharp constant for the Gagliardo-Nirenberg inequality ∥f∥4L4 ≤ c ∥∇f∥3L2 ∥f∥L2, a valuable result for understanding the dicotomy global well-posedness and blow-up, that we then show. Finally, in Chapter 5, we switch to the study of the global solution's assymptotic behaviour, showing that, below the threshold mass-energy of the associated elliptic equation, the global solutions of (1) scatters.CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível SuperiorporUniversidade Federal de Minas GeraisPrograma de Pós-Graduação em MatemáticaUFMGBrasilICX - DEPARTAMENTO DE MATEMÁTICAMatemática – TesesEquação de Schrödinger – TesesInterpolação– TesesEspalhamento – TesesEquação de Schrödinger não-linearEspaço de energiaInterpolaçãoBoa-colocaçãoExplosãoEspalhamentoBoa colocação e comportamento de soluções no espaço de energia para a equação de Schrödinger não-linear 3D cúbicaWell-posedness and behavior of solutions in the energy space for the non-linear 3D cubic Schrödinger's equationinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/masterThesisinfo:eu-repo/semantics/openAccessreponame:Repositório Institucional da UFMGinstname:Universidade Federal de Minas Gerais (UFMG)instacron:UFMGORIGINALalan-dissertação-final.pdfalan-dissertação-final.pdfapplication/pdf1268940https://repositorio.ufmg.br/bitstream/1843/61123/1/alan-disserta%c3%a7%c3%a3o-final.pdf7cd9e78ac5b5ea623ac2a52a00780a19MD51LICENSElicense.txtlicense.txttext/plain; charset=utf-82118https://repositorio.ufmg.br/bitstream/1843/61123/2/license.txtcda590c95a0b51b4d15f60c9642ca272MD521843/611232023-11-20 08:57:28.182oai:repositorio.ufmg.br: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ório InstitucionalPUBhttps://repositorio.ufmg.br/oaiopendoar:2023-11-20T11:57:28Repositório Institucional da UFMG - Universidade Federal de Minas Gerais (UFMG)false
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