Well-posedness For A Class Of Coupled Klein-Gordon Equations And Nonlinear Schr(?)dinger Equations

This paper carries out well-posedness studies of a coupled system of nonlinear Klein-Gordon equations with strongly damping terms and two classes of nonlinear Schrodinger equa-tions. The parts of Schrodinger equations contain nonlinear second-order Schrodinger equations with general source term and nonlinear fourth-order Schrodinger equations with radial data. In Part 1, we give the physical background and research progress of nonlinear Klein-Gordon equa-tions and Schrodinger equations and summarize the work of this paper.In Part 2, we concern with the global well-posedness of Cauchy problem for a coupled sys-tem of nonlinear Klein-Gordon equations with strongly damping terms. Klein-Gordon equation is a very important nonlinear evolution equation in the area of mathematical physics. Firstly, by constructing the energy functionals, establishing variational structures we obtain the local exis-tence of solutions. Moreover,using energy estimations and boundness of the norms we get the global existence of solutions and combining multiplier method we have the asymptotic behavior of the global solution. Finally, for the strong damping equations, we redefine the auxiliary func-tions combining the potential well theory, concavity method, embedding theorem and properties of the norms to prove a blow up in finite time result.In Part 3, we investigate the sharp conditions for the Cauchy problems of the nonlinear 2-dimensional second-order Schrodinger equations and N-dimensional fourth-order Schrodinger equations with radial data. The Schrodinger equation is the most fundamental equation in quan-tum mechanics. Therefore, it is necessary to study these issues. For second-order equations,by constructing the variational structures and the functional spaces we analyze the properties of the Nehari manifold of the solutions. Moreover, we give the sharp conditions of global ex-istence and blow up results by potential well theory and concavity method. For fourth-order equations, because the structure of the potential wells of fourth order nonlinear Schrodinger equations has a more complex structure than the second-order Schrodinger equations,so we chose a more proper functionals space H2 as its initial value space. Then the remainder of proof for the fourth-order cases is the same as the proof for second-order cases.