On The Study For Some Nonlinear Wave Equations

In this paper, the nonlinear Schrodinger equation with different power nonlinearities and the nonlinear wave equations of Kirchhoff type with nonlinear damping and source terms are considered:In Chapter 1, some backgrounds, main tools are reviewed, and some important results concerned are presented.In Chapter 2, the nonlinear Schrodinger equation with different power nonlinearities is considered. By constructing a type of cross-constrained variational problem and establishing so-called cross-invariant manifolds of the evolution flow, we derive a sharp threshold for global existence and blowup of the solution to the Cauchy problem.In Chapter 3 and 4, the nondegenerate and mildly degenerate Kirchhoff type equations with nonlinear damping and source terms are considered. We obtain the sufficient condition for the blow-up of the solutions by using the so-called concavity method. At the same time, we remark that the nonlinear damping terms have stronger effect than the linear damping terms with regarding the blow up results.In one word, for the nonlinear Schrodinger equation with different power nonlinearities, we get the blow-up and sharp condition of global existence of solutions, by using the variational methods introduced by J.Zhang; for the Kirchhoff-type equation with nonlinear damping and source terms, we obtain the sufficient conditions of the blow-up of solution in the cases of nondegenerate and midly degenerate, by using the concavity methods.