On The Klein-Gordon-Zakharov Equations In Three Space Dimensions

Nonlinear wave system is one of the most attracting research fields in mathematical physics. On the one hand it delineates some abstruse laws and rules of motion in modern physics, and on the other hand, as one of the most important partial differential equations, it is one important part of the core mathematics. Classical nonlinear wave equations mainly include the Korteweg-de Vries equation, the nonlinear Schrodinger equation and the nonlinear Klein-Gordon equation etc. Korteweg-de Vries equation is a typical solitary wave model, and nonlinear Schrodinger equations as well as nonlinear Klein-Gordon equations are major models in quantum mechanics.In the recent twenty years, a series of important advances are achieved on the mathematical studies around the above three models. Especially for their typical properties such as the local well-posedness of the initial value problems, blowingup properties of the solutions in a finite time and their dynamical behavior, existence of the global solutions and their asymptotic behavior, existence of the standing waves and their stability, plentiful and substantial results are got. As the representative persons who accomplished these studies, mathematicians such as W.A.Strauss, J.Ginibre, T.Cazenave, F.Merle, Y.Tsutsumi, T.S.Li and B.L.Guo play a symbolic role in the modern advance of PDE and the core mathematics.The Klein-Gordon-Zakharov equations are an important nonlinear wavemodel that is noticed in the recent five years. It is a coupled mathematical physics system which describes the interaction of the Langmuir wave and the ion acoustic wave in a plasma and so on. This system is a nonlinear coupled equations which are formed by a Klein-Gordon equation and a classical hyperbolic wave equation according to a coupled means of the Za-kharov equations. Besides the explicit meaning in physical background, it has typical feature in Mathematics. In [23], Ozawa, Tsutaya and Tsutsumi studied the Cauchy problem of the Klein-Gordon-Zakharov equations with different propagation speeds in three space dimensions. In terms of the correlated theory of the homogeneous Sobolev space of negative index and the discrepancy between the propagation speeds in the Klein-Gordon-Zakharov equations, they obtained the local well-posedness for the Cauchy problem in the energy space by using a harmonic analysis method and a contraction method. In addition, by combining this result and the energy conservation law, they obtained the unique global solutions for the Cauchy problem of the Klein-Gordon-Zakharov equations in the energy space for small initial data. Moreover, in [2,24], they got the global existence of small amplitude solution for the Cauchy problem of the Klein-Gordon-Zakharov equations in the case of c = 1.In the present paper, combining the local well-posedness for the Cauchy problem of the Klein-Gordon-Zakharov equations and the related theory of the homogeneous Sobolev space of negative index, we study the Klein-Gordon-Zakharov equations with different propagation speeds in three space dimensions under the framework of variational calculus, we first obtain the existence of the standing waves for the Klein-Gordon-Zakharov equations under the case of non-radial symmetry and radial symmetry by constructing two kinds of different constrained variational problems. We next get that thesolution for the Cauchy problem of the Klein-Gordon-Zakharov equations blows up in a finite time when the initial energy is negative by applying the potential well argument and the concavity method. We then derive out the sharp condition of global existence for the Cauchy problem of the Klein-Gordon-Zakharov equations by using scaling argument. Finally, in the light of the harmonic analysis method and the lower semi-continuity of weak limit, we prove the instability of the standing waves for the Klein-Gordon-Zakharov equations under the case of radial symmetry and non-radial symmetry.In introduction, first of all, we present the physical background and some known results on the Klein-Gordon-Zakharov equations. Secondly, we recall some known results on the classical nonlinear Klein-Gordon equation and Schrodinger equation as well as the Zakharov system. Finally, we present the main results of the present paper.In chapter 2, we first study the local well-posedness for the Cauchy problem of the Klein-Gordon-Zakharov equations with different propagation speeds in three space dimensions in the energy space, and obtain the unique existence of the solutions for the Cauchy problem on a maximal time interval in the light of [23]. We next introduce the expression form of the definition and the norm about the homogeneous Sobolev space of negative index and radially symmetric Sobolev space. At last, we construct proper functionals and manifolds in view of the distinctive feature of the Klein-Gordon-Zakharov equations.In chapter 3, we study the standing waves of the Klein-Gordon-Zakharov equations under the case of non-radial symmetry and radial symmetry, respectively. First of all, we construct two proper constrained variational problems. Next, according to the characteristics of the ground state andthe local theory, we prove the existence of the non-radial symmetric standing waves by variational calculus. Then, by solving a constrained varia-tional problem, which is equivalent to the constrained variational problem correlated to the radially symmetric standing waves, we obtain the solution of the constrained variational problem. Finally, by Lagrange multiplier method, we prove that the solution of the constrained variational problem is the solution of the nonlinear elliptic equations corresponding to the Klein-Gordon-Zakharov equations. Thus the existence of the radially symmetric standing waves is established.In chapter 4, we study the blowup of the solution for the Cauchy problem of the Klein-Gordon-Zakharov equations. Based on the characteristics of the ground state and the existence of the non-radially symmetric standing waves, we obtain the solution of the Cauchy problem blows up in a finite time when initial energy is negative by using the potential well argument and the concavity method.In chapter 5, we study the global solutions for the Cauchy problem of the Klein-Gordon-Zakharov equations. According to the existence of the standing waves with ground state under the case of non-radial symmetry, using the result in chapter 4, we derive out a sharp condition of global existence and blow up for the Cauchy problem. Moreover, by using scaling argument, we answer the question of how small the initial data are, the global solutions of the Cauchy problem exist.In chapter 6, we prove the instability of the standing waves for the Klein-Gordon-Zakharov equations under the case of non-radial symmetry and radial symmetry, respectively. Firstly, we obtain the strong instability of the standing waves under the case of non-radial symmetry by applying potential well argument and the concavity method. Secondly, by usingthe original definition of stability, we show that the standing waves for the Klein-Gordon-Zakharov equations under the case of radial symmetry are always instability, regardless of the solutions for the Cauchy problem of the Klein-Gordon-Zakharov equations exist locally or globally. In the end of this chapter, we compare the two type of standing waves and their instabilities under the case of non-radial symmetry and radial symmetry for the Klein-Gordon-Zakharov equations and obtain that the standing waves under the two cases are different. The instability of the standing waves under the case of non-radial symmetry is initiated by blow up in a finite time, yet, the instability of the standing waves under the case of radial symmetry has nothing to do with the solutions blow up in a finite time or exist globally.