Study On The Well Posedness Of Solution For The Schr(?)dinger Equations In The Sense Of Non Square Integrable Initial Values

In this paper,the well posedness of local and global solutions of second-order and fourth-order Schr(?)dinger equations are studied in a space where the initial value belongs to non square integrable space L~por L~p(p=2).During the proofs,we mainly use the fixed point theorem,the homogeneous and non-homogeneous Strichartz estimates and the initial value decomposi-tion method.The purpose is to generalize the well posedness conclusion of the second-order Schr(?)dinger equation under different nonlinear terms and supplement the well posedness re-search of the fourth-order Schr(?)dinger equation in the sense of non square integrable initial value.In Chapter one,some research background of the Schr(?)dinger equation are briefly re-viewed,and the research ideas with starting points of this paper are given in detail.In Chapter two,the Cauchy problem of second-order Schr(?)dinger equation with Hardy nonlinear term under the condition that the initial value belongs to non square integrable space L~pis studied.In the section of preparatory works,two kinds of Strichartz estimates based on the Lorentz space and the dependence of the solution on the initial value in the sense of L~2initial value are given.Then,when the initial data belongs to the initial value decomposition space which is defined in this chapter,the local well posedness is proved by using the fixed point theorem,and under the some parameter conditions,the maximum existence interval of local solution can extended to infinity.Finally,through the embedding properties of initial value decomposition space and L~p,the well posedness of global solution with the initial value belongs to L~pwas proved.In Chapter three,the Cauchy problem of the fourth-order Schr(?)dinger equation with Hartree nonlinear term under the condition that the initial value belongs to the non square integrable space L~pis studied.The research method is similar to that in Chapter two.The difference is that the initial value decomposition form of the problem is different,which leads to the different forms of its decomposition space and Strichartz estimates,and the expression of the dependence of the solution on the initial value in the sense of L~2initial value is also dif-ferent.In this chapter,the corresponding proofs are given for these different aspects.Finally,the conclusions for the well posedness of the local global solutions are also obtained.In Chapter four,in the sense of Fourier transform,the Cauchy problem of one-dimensional fourth-order Schr(?)dinger equation is studied when the initial value belongs to non square in-tegrable space L~p.Based on the decomposition properties of L~p,in this chapter,the Strichartz estimates based on L~pspace and the dependence of the solution on the initial value in the sense of L~2initial value are established,then we prove the well posedness of local and global solutions.