O problema de Cauchy para a equação de Schrodinger não-linear não-local

AUTOR(ES)

Roger Peres de Moura

DATA DE PUBLICAÇÃO

2005

RESUMO

ln this work we establish some properties of the nonlocal nonlinear Schrodinger equation (NLSNL). First of alI, we present a preliminary chapter with notations and basic theory used to establish our results; that part also seeks to facilitate the reading of this work. Soon afterwards comes the main result: local welI-posedness for the initial value problem (the Cauchy problem or lVP) for the NLSNL equation with initial data in real Sobolev spaces of index larger than three and a half; the method of proof alIows to es- tablish that the data-solution map is smooth. ln the folIowing chapter we proved that previous result for the intermediate nonlocal nonlinear Schrüdinger (lNLSNL), which is more general than the NLSNL equation. After that we establish local welI-posedness for the NLSNL equation in weighted Sobolev spaces. ln another chapter the ill-posedness issue is discussed: we established that one cannot obtain local welI-posedness, in Sobolev spaces of negative index, for the lVP associated to NLSNL equation through a iterative Picard method; as a consequence, the data-solution map is not smooth in those spaces. We also proved, making use of a Pohozaev s identity, the no-existence of standing waves solutions for the NLSNL equation. We concluded with a chapter where we exhibited some interesting problems mainly related to the NLSNL equation and possible difficulties to be faced in an eventual attempt of solving them

ASSUNTO(S)

schrodinger schrodinger equation problemas de cauchy problems partial differential equations equações cauchy equações diferenciais parciais

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