Well-posedness And Blow-up Of Two Kinds Of Nonlinear Schr?dinger Equations

The nonlinear Schr?dinger equation is always one of research hotspot of the partial differential equation.Especially in recently 50 years,with the introduction of harmonic analysis technique and concentration-compactness method,mainly by the important work of a famous mathematicians Bourgain,Tao,Kenig and Merle,so that the study of this equation has made great progress.In this paper,we mainly study the well-posedness and the blow-up of two kinds of nonlinear Schr?dinger equations,that is the sharp well-posedness of the Cauchy problem for the fourth order nonlinear Schr?dinger equation and the well-posedness and blow-up of the Cauchy problem for the second order nonlinear Schr?dinger equation.The difference of the nonlinear Schr?dinger equation mainly demonstrates in the structure of the nonlinear term,thus behaving different well-posedness and blow-up.In Chapter 2,we investigate the Cauchy problem for the fourth order nonlinear Schr?dinger equation i(?)tu +(?)x4u=u2,(t,x)?[0,T]×R.The known Bourgain space Xs,b can not be suitable for the research of the lower regularity index.To overcome this difficulty,according to the special property of the nonlinearity u2,we choose a proper weight function and introduce a modified Bourgain space Xs,b.Thus we can get a sharp regularity index by establishing some refinement bilinear estimates and taking full advantage of the Young inequality and the Holder inequality.That is we prove that the equation is locally well-posed in HS(R)as s?-2 and ill-posed as s<-2 in the sense that the solution map is discontinuous.In Chapter 3,we consider the well-posedness and the blow-up of the Cauchy problem for the second order nonlinear Schr?dinger equationFirstly,we prove the local well-posedness of the Cauchy problem with initial data in L2(Rn)via the contraction mapping principle.Then,by the test function method and the weak solution,we get an important integral inequality.Therefore,under a suitable condition on the initial data,we conclude that the L2-norm of the solution can blow up in finite time although the initial data are arbitrarily small when 1<P1,p2<1+4/n.As a by-product,we also obtain an upper bound of the maximal existence time of the solution.In Chapter 4,we further study the Cauchy problem on the second order nonlinear Schr?dinger equation based on the previous chapter.Using the test function method and the weak solution analogous to Chapter 3,by choosing the initial value of different conditions,we obtain the global well-posedness and blow-up results of this Cauchy prob-lem in H1(Rn).Specifically,when the nonlinear indexes satisfy 1<p1,p2<1+4/n-2,we prove the local well-posedness of the Cauchy problem in H1(Rn).In addition,if 1+4/n?p1,p2<1+4/n-2,we prove the global well-posedness with small initial data in H1(Rn).While under a suitable condition on the initial data and min(p1,p2)<1+4/n,we prove that the H1-norm of the solution would blow up in finite time although the initial data are arbitrarily small.Meanwhile,we give a large initial data blow-up result when p1,p2<1+4/n-2.Finally,we show the non-existence of the local weak solution for some Hm-data with m=0,1 as max(p1,p2)>1+4/n-2m.