A Study On The Solution Of Schr(?)dinger Equation With Subcritical Nonlinearity

Nonlinear science is an interdisciplinary comprehensive science that has gradually formed in the past 50 years.Nonlinear science has been involved in various fields,which greatly affects the logical system of modern science.Nonlinear differential equations have become the central content that needs to be studied in various disciplines and fields among them.Whether in scientific theory or in practical applications,nonlinear partial differential equations have very important practical applications.It is usually used to study problems in the fields of mechanics,engineering sciences,life sciences;and epidemiology.In nonlinear partial differential equations,the nonlinear Schrodinger equation is a very important model,which has very broad prospects in various scientific fields.By using harmonic analysis,in this thesis we mainly study the following initial value problem of Schrodinger equations with subcritical nonlinearities(?)The above equations have important applications in various fields such as nonlinear optics,plasma physics,and fluid mechanics.We show the existence of global solutions of the large initial problem and time decay estimates of the solutions under different value ranges of p,it has specific use value in quantum mechanics,optics and other disciplines.This thesis is composed of four parts.In the first part,we first show the research background of this thesis,and briefly review the relative research progress of nonlinear Schrodinger equations.Then we summarize the main work in this thesis.In the second part,we introduce the basic notation used in this thesis,as well as the basic theorems and some common inequalities used in the thesis.In the third part,we mainly consider the Schrodinger equations with subcritical nonlinearities for low regularity data.We show the existence of global solutions of the large initial problem and time decay estimates of the solutions.Moreover,we study the asymptotic behavior of the solutions.In the fourth part,we study two situations of the initial problem if ?(t)=?/(1+t)(p-1)When the initial data satisfies v₀?H^0,?(R),we use harmonic analysis method and factorization formula to study the existence of the global solution to the nonlinear Schrodinger equations and the L^? time decay estimates of the solutions.When the initial data satisfies v₀ ?H^0,1(R)?H¹(R),we use the energy method to investigate the existence of solutions of the initial problem.Finally,we use the harmonic analysis method and the Young's inequality to show the L² time decay estimates of the solutions.