Wellposedness And Scattering Of The Generalized Boussinesq Equation And Schr(?)dinger-Boussinesq System

This paper is concerned with the Schr(?)dinger-Boussinesq system,the generalized Boussinesq equation and the generalized Schr(?)dinger-Boussinesq system.In Chapter 2,we obtain local and global wellposedness,finite-time blowup of the Schr(?)dinger-damped Boussinesq system.We use Strichartz estimates of the classical Schr(?)dinger equation and the linear estimates of the damped Boussinesq equation obtained in Wang et al.[1]to get the local wellposedness.Then,we ultilize the method used in Wang et al.[1]to discuss the global wellposedness and finite-time blowup.Meanwhile,the one dimension fractional Schr(?)dinger-damped Boussinesq system is researched in the same manner.In Chapter 3,we derive that the one dimension Schr(?)dinger-Boussinesq system with(u,v.(-?xx)-1/2vt)is local wellposed in Hs × Hs × Hs-1,s≥-1/4,thus improve the work of Farah[4].The main proof method is inspired by the work of Kishimoto-Tsugawa[2],Kishimoto[3].In Chapter 4,we study wellposedness and scattering of the generalized Boussinesq equation.By using the Strichartz type estimates obtained in Gustafson et al.[5],we derive the local wellposedness and small initial data scattering that is parallel to those of the nonlinear Schr(?)dinger equation.Then,we utilize the method of Wang et al.[1]to obtain the global wellposedness and finite-time blowup.Finally,inspired by Dodson-Murphy[6],we obtain large radial initial data scattering of the defocusing case for d≥3.By the same method,we obtain the local and global wellposedness,finite-time blowup and small initial data scattering of the generalized Schr(?)dinger-Boussinesq system in Chapter 5 and thus remove the damping term-αΔnt,α>0 of the Schr(?)dinger-damped Boussinesq system in Chapter 2.