A Study On The Solution To The Nonlinear Schrodinger Equation With Dissipative Nonlinearities

The nonlinear partial differential equation is one of the important branches in modern mathematics and it has a lot of applications in the fields of nature science,engineering,economic management,etc.The nonlinear Schrodinger(NLS)equation is an important nonlinear partial differential equation,its related theories are attracting more and more attention from mathematicians and physicists.Meanwhile,The NLS equation provides a multifaceted clue for helping people study its properties.In this paper we mainly discuss the time decay estimates of the solutions to the NLS equation with strong dissipative nonlinearities by using some mathematical tools in harmonic analysis.We divide this paper into three parts.In Part ?,we make an introduction to the background of the NLS equation and give an outline of current progress in the study of the NLS equation.In Part ?,we show the basic concepts and some classical results that we will use.Finally in Part ?,we focus on the solution of the strong dissipative NLS equation and show time decay estimates of solutions.First,we prove the global existence of solutions to this NLS equation by using the energy method.Then we combine harmonic analysis,factorization formulas and Young's inequality to get the time decay estimates.Moreover,we obtain a better time decay estimate of solutions to the strong dissipative nonlinear Schrodinger equation under some assumptions of the damped nonlinearities.