Asymptotic behavior for inhomogeneous nonlinear Schrödinger Equation

• Autor(a) principal: Mykael de Araújo Cardoso
• Data de Publicação: 2020
• Tipo de documento: Tese
• Idioma: eng
• Título da fonte: Repositório Institucional da UFMG
• Texto Completo: http://hdl.handle.net/1843/33567
• topic: Well-posedness; Nonlinear Schrödinger equation; stability; blow-up of the ccritical norm; Matemática - Teses.; Equações diferenciais parciais.; Problemas de valor inicial; Schrodinger, Equação de

In this thesis we investigate some questions about the long-time behavior of the solutions for the initial value problem (IVP) associated to the inhomogeneous nonlinear Schr\"odinger (INLS) equation $$ i \partial_t u + \Delta u + \kappa|x|^{-b} |u|^{2\sigma}u = 0, $$ where $\kappa=\pm 1$ and $\sigma, b>0$. Among them, (a) stability of standing waves for focusing $L^2$-subcritical INLS equation for which we give an alternative proof for the result of De Bouard and Fukuizumi[9] (b) local well-posedness for the intercritical INLS equation in $\dot H^{s_c}(\Real^N)\cap \dot H^1(\Real^N)$; (c) critical norm concentration for finite-time blow up solutions; (d) blow-up of the critical norm for solutions with radially symmetric initial data in $\dot H^{s_c}(\Real^N)\cap\dot H^{1}(\Real^N)$, inspired by the idea of Merle and Raphäel [52].