Boa colocação e comportamento de soluções no espaço de energia para a equação de Schrödinger não-linear 3D cúbica

Abstract

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.