Study On The Long Time Behavior For Some Nonlinear Partial Differential Equations

In this doctoral dissertation,we devote to study the stability theories of the solitary wave solution,well-posedness and scattering theory for the nonlinear dispersion equa-tions and the long time behavior of the two-fluid model.It is made up of five chapters.The first chapter is a summary,which is divided into five sections;the first section is the research background and progress of this paper;the second,third,fourth and fifth sections respectively provide the background and research progress of the model studied in this paper,as well as the main conclusions obtained.In Chapter Two,we consider the instability of the solitary wave solution for the generalized Boussinesq equation.The generalized Boussinesq equation is written as(?)(t,x)? R × R,with 0<p<?.This equation has the traveling wave solutions ??(x-?t),with the frequency ??(-1,1)and ?? satisfying-(?)xx??+(1-?2)??-??p+1=0.Bona and Sachs(1988)proved that the traveling wave ??(x-?t)is orbitally stable when 0<p<4.p/4<?2<1.Subsequently,Liu(1993)proved the orbital instability under the conditions 0<p<4,?2<q/4 or p?4,?2<1.For the only remaining problem,that is,the degenerate case 0<p<4,?2=p/4.We prove the orbital instability of the solitary wave solution.In Chapter Three,we consider the instability of the solitary wave solution for the generalized derivative nonlinear Schrodinger equation.The generalized derivative non?linear Schrodinger equation is written as where 1<?<2.The equation has a two-parameter family of solitary wave solutions of the form u?,c(t,x)=ei?t+ic/2(x-ct)-i/2?+2(?)??,c2???,c(x-ct).The stability theory with the frequency region |c|<2(?)was studied by Liu,Simpson and Sulem(2013)and Guo,Ning and Wu(2018),etc.In this chapter,we prove the instability of the solitary wave solutions in the endpoint case c=2(?).In Chapter Four,we consider the well-posedness and scattering theory of the non-linear Schrodinger equation.The nonlinear Schrodinger equation is written as i(?)tu+?u=?|u|pu,(t,x)? R1+2,with ?=±1 and p>0.We prove that for a radial function f ?H?(R2)(s0<sc),with the support away from the origin,there exists an incoming and outgoing decomposition f=f++f-,such that the radial initial data in the outgoing part f+(or incoming part f-)leads to the local well-posedness and small data scattering in the forward(or backward)time.In Chapter Five,we construct an alternative proof for the long-time behavior of large-data classical solutions to the two dimensions semi-dissipative Boussinesq equa-tions without thermal diffusion on a bounded domain subject to the stress-free boundary conditions,which was previously studied by Doering,Wu,Zhao and Zheng(2018).To demonstrate the effectiveness of the new approach,we study the long-time behavior of large-data classical solutions to the initial-boundary value problem of a related model with density variance and subject to the no-flow boundary condition.