Boa e má colocação para a equação de Schrödinger não-linear

Abstract

The purpose of this work is to study well-posedness and ill-posedness of the initialvalue problem (IVP) for the nonlinear Schrödinger equation. In the first chapter, we present some important results of Harmonic Analysis and we study the Sobolev spaces Hs(RN), which provide the fundamental basis of our work. In the second chapter, we will show, in the subcritical case, local and global well-posedness of the (IVP) associated to equation (2) with data in L2(RN). Furthermore, in the critical case, we will show local well-posedness and, on special case, global well-posedness. Finally, in the chapter 3, we study the (IVP) associated to equation (2) in Hs(RN) for negative indices s. We will show ill-posedness for s (N/2 - 2/ , 0) and ill-posedness of the (IVP) with the Delta function as initial datum.