Initial Boundary Value Problem of Several Nonlinearity Schr(?)dinger Types Equations
|Keywords||System of nonlinear equations Galerkin method Initial boundary value problem|
In this article, we consider initial boundary value problem of several nonlinearity Schrodinger types equations .Under the proper condition, the global solution existence of these problems are proved. In .the following nonlinear coupling Schrodinger-Klein-Gordon equations isgiven:This is the first researching work about this problem in the world.The author research into the cauehy problem of this equations in R3 and acquire the global existence and uniqueness of solution.In[4、14]deal with a class nonlinear Schrodinger-Boussinesq types equations and obtain the global adaptive result of initial boundary value problem about the equations with magnetic field effect. This research is the newest work about this problem in the world.In . Brezis-Gallont research into the cauehy problem of generic Schro dinger equation in R2 and particular investigate the nature and solution of the equation ,as follow:In [10,11],using Galerkin method, Tsutsuini extented their results. From the view of mathematical. these equations or equation are important for mordern physics [6、7、8、 9]. Based on the previous study made by pioneering scholars, we carry over the study work forwarding.In detail,on Chapter two, we study the following new model:Using Galerkin method,we get the following theorem.Theorem Let and or set Under assumption one of these conditions hold:（i）β≥0,η≥0,q,λare real and（ii）β≥0,η, q,λare real and（iii）β< 0,η< 0, q,λare real and（iv）β< 0,η≥0, q,λare real andor set Under assumption one of these conditions hold:（i）β≥0,η≥0,q,λare real and d（Ω）（ii）β≥0,η,q,λare real andη＜0,（iii）β< 0,η≥0, q,λare real and（iv）β< 0,η≥0, q,λare real andwhere d（Ω） representing intervalΩmeasure. Then problem （3） exist the globel weak solution such that: In Chapter 3, we consider a class initial bounded problem of nonlinear Schrodinger-Boussinesg type equation system:Whereγ,λis constant,βis a real.ε|→= （ε1,ε2,ε3） is the three dimension compound unknown function of vector.n（x. t）,φ（x, t） are unknow functon.Ω∈R2 has smooth bounded , and q（s）,f（s）,a（x） are known function. Using Galerkin method and senligroup of operator method and compactness principle, we studied global weak solution existence of the problem and obtain the following results:Theorem Let ;or set R2. Under assumption one of these conditions hold:（i）β≥0,η≥0, q.λare real and（ii）β≥0,η,q,λare real andη< 0,（iii）β＜0,η, < 0, q.λare real and（iv）β< 0,η≥0, q,λare real andor set R1 , Under assumption one of these conditions hold:（i）β≥0,η≥0, q,λare real（ii）β≥0,η, q,λare real andη< 0,（iii）β< 0,η≥0,q,λare real and（iv）β< 0,η,≥0,q,λare real andwhere d（Ω） representing intervalΩmeasure. Then problem （4） exist the solution such that:In Chapter 4,we consider the folowing initinal bounded problem of nonlinear Schrodinger equations with magnetic field effect:whereΩ∈R2 has smooth bounded such that:Using similar method as Chapter 3, we proved the existence and uniqueness of problem （5） . We obtain the following results:Theorem Under assumptions （6）-（8） and such that Then the problem （5） existenceuniqueness solution u|-（x,t） such thatOur research extend results made about Schro dinger eciuation and equations to a certain extent.