Global Well-posedness And Limit Behavior Of The Solutions To A Class Of Nonlinear Dispersive Wave Equations

In this paper the global well-posedness and limit behavior of the solutions to the nonlinear dispersive wave equations are studies.Get the global existence and uniqueness of solution to the nonlinear dispersive wave equations for u₀∈L²(R). According to the energy estimate,we result the local well-posedness for u₀∈L²(R). The strong and weak limit behavior of solutions to the nonlinear dispersive wave equations are gotten.There are five sections in this paper.The first section,we introduce the background and actuality and summarize the main result.The second section,we will introduce the basic theory,basic concepts needed in the study's.The third section,we use the energy estimate and the usual extension theory,the global well-posedness of the dispersive DP equation is proved;The strong and weak limit behavior of solutions to the nonlinear dispersive wave equations are gotten.The forth section,studies the weak solution,entropy weak solution for nonlinear dispersive wave equations.By using viscous approximations and a priori estimates, we prove the existence of at least one weak solution,satisfying a restricted set of entropy inequality in the class L²(R)∩L⁴(R),and in L¹(R)∩BV(R) existing entropy weak solution.The fifth section,we study the well-posedness of the b-family equations,we mainly used the the energy estimate and the usual extension theory.