The Research On Global Well-posedness For Some Classes Of Nonlinear Systems

In this thesis,we study the initial boundary value problems for two classes of coupled reaction diffusion systems and a class of nonlinear Cahn-Hiliard equations and the initial value problem for a class of strong damping term Boussinesq equations and a class of fourth order nonlinear Schr(?)dinger equations.Depending on suitable initial data,we obtain the global existence and nonexistence solution of different problems under the conditions of nonlinear term exponents,and analyze the development of the solutions.We analyze the global well-posedness of two class of reaction-diffusion systems with coupled source terms under the conditions of low,critical and high initial energy level.These two systems are mostly used to describe the heat transfer in physics,the conversion of substances in chemistry and the multiplication of species in biology and so on.Through using the symmetry of coupling source terms,we skillfully obtain the energy functional and Nehari functional,and establish the framework of potential well theory.By using the Galerkin method and the concave function method,we obtain the global well-posedness of solutions in low and critical energy.In particular,we establish the comparison principle of reaction-diffusion systems.By the variational method,we obtain the global existence and finite time blow up solutions of reaction-diffusion systems under high initial energy level.We study the initial boundary value problem for the Cahn-Hiliard equation.This model is used to describe the growth and diffusion of population which is sensitive to time period factors.The dispersion term emphasizes the differentiability of the space,and the anisotropic term reflects the multi-directional of the space.By using a large number of analytical techniques,the fourth order dispersion term and the general anisotropic term are dealt with from the multidimensional point of view.We improve the estimation accuracy of potential well depth,and obtain the range of dispersion term and anisotropic term from the range of Nehari functionals.Furthermore,we give the global existence and non existence of solutions under low energy.We study a class of sixth order Boussinesq equation in multidimensional space.The Boussinesq equation is mainly used to describe the propagation of water waves,such as the propagation of small amplitude waves in shallow water.By using the Fourier transformation,we deal with the nondecreasing term of the equation,and introduce the energy functional andNehari functional.We give the global existence solution under the low energy level condition.However,strong damping term of the equation can not be used the traditional method of concave function directly to solve.Therefore,by improving the concave function method,introducing new auxiliary function,and further exploring the properties of the Nehari flow,we get the conditions for nonexistence of global solutions.Then we give the threshold condition for the existence and nonexistence of the solution.Finally,we analyze the relation between the strong damping coefficient and the explosive crack.We study the Cauchy problems of a class of fourth order Schr(?)dinger equation.Using the structure of the equation,we propose the potential well,and then we obtain the energy conservation,mass conservation and get the global existence solution.In particular,we introduce radial auxiliary function,and analyze the possibility of blow up in finite time.