Initial Boundary Value Problems For Two Kinds Of Nonlinear Evolution Equations (Systems) With Double Dispersion

Nonlinear wave equation is an important research field of partial differential equation.In physical problems,the three factors of nonlinearity,dispersion and dissipation affect the wave propagation in elastic rod.Among them,the nonlinearity makes the wave front steepen or even break up,the dispersion and dissipation can reduce the slope of the wave front and prevent the wave from breaking up,so that the internal wave of elastic rod will get the final steady state.What’s more,this paper is divided into three chapters to discuss the initial boundary value problems of nonlinear evolution equations(Systems)with double dispersion.In the first chapter of this paper,we introduce the background of nonlinear evolution equation(Systems)with double dispersion.In the second chapter,we study a class of nonlinear evolution systems with double dispersion.By using the Galerkin method,the existence and uniqueness of global strong solution for the problems(2.1)-(2.3)are proved.In the third chapter,we prove the existence of global weak solution and the existence and uniqueness of global strong solution to a class of nonlinear evolution equation with double dispersion by the potential well method.