Blow-up Of Solutions To A Class Of Nonlinear Schr(?)dinger Equations

With the continuous development of science and technology and basic theory of mathematics,people pay more and more attention to nonlinear problems.Nonlinear science has gradually become one of the important research directions in modern science.Nonlinear Schrodinger equation is one of the most important and universal nonlinear models in modern science,which originated from applied mathematics,physics and other applied disciplines.It can be used to describe various nonlinear waves in physics,such as laser beam propagation in media with refractive index related to wave amplitude,ideal fluid water wave on free surface,plasma wave,etc.In this paper,we study the following nonlinear Schrodinger equation:i?tu=-Δu+i(-t)a(p-1)|u|p-1u,where p>1,(n-2)(p-1)≤4,a≥0 is a real number,(t,x)∈(-∞,0)×Rn,u=u(t,x)is an unknown complex value function.This paper consists of four parts.In the first part,we introduce the research background of this paper,briefly describe the research status of blow-up solutions of nonlinear Schrodinger equation at home and abroad,and summarize the main research content of this paper.In the second part,we give the basic symbols,fundamental concepts and some important inequalities used.In the third part,we study the existence and uniqueness of global solutions of the backwards equations studied in this paper.In the fourth part,we use the energy method and the compact method to prove the blow-up of the solution of the equation.This section consists of three parts.First,we construct an approximate solution of the equation studied.The idea is to construct a explicit function Φ(t,x)=(C(-t)a(p-1)+1+φ(x))-1/(p-1),where C=(p-1)/(a(p-1)+1),(t,x)∈(-∞,0)×Rn.And the function Φ satisfies the ordinary differential equation of Φt=(-t)a(p-1)|Φ|p-1Φ with a series of assumptions about φ,such that ‖Φ‖L2(Rn)→∞ as t→0-.Secondly,we use the energy method and some important inequalities to estimate the error term.Finally,a solution of the equation is found by using compactness method,and this solution tends to the approximate solution Φ.Then we prove the final blow-up result by using the previous estimate.