Unique Solvability And Blow-up In The Energy Space For KGS-type System

As one of the important models in quantum physics, Klein-Gordon-Schrodinger (KGS) system which describes the interaction between the nucleon field and meson field in an atomic nucleus not only reveals the most profound motion law of parti-cles in the modern physics. Meanwhile, as a significant class of mixed type partial differential equations, it has become one of the most important directions that lead modern mathematical research. In recent years, scholars have carried out exten-sive studies to this system and obtained many substantial achievements. With the deepening of the research, various KGS systems in sophisticated environments have been derived and caused widespread concerns again. However, most of theses given results focus on the attractor and energy decay of the dissipation systems. This the-sis is concerned with the high-order nonlinear KGS-type systems. We will mainly explore the posedness in the energy space and blow-up attributes. This dissertation is divided into five sections.First, a physical description and some basic research status as well as the main issues and results of this study are presented for KGS system in Introduction. At the end of this section, by using a viscosity method, Lp-Lq estimates and compactness arguments, we prove the existence of the energy solutions for the KGS-type system with higher order nonlinear terms, and point out the solution is global for the system with Yukawa momentum.Second, KGS-type system with dissipation term is considered in Chapter 2. By introducing an integral-type auxiliary function with respect to the time, with the help of the energy estimate, the uniqueness of the energy solution for dissipative KGS-system is shown. Furthermore, the same procedures are applied to prove the continuous dependence of the solution on the initial data.Third, the conserved KGS-type system in R2+1. Schrodinger equation with perturbed Laplace oprator is concerned in Chapter 3. A Kato-type smooth estimate of Schrodinger equation under the nonlinear effect of the potential field in H1 is derived, here the potential field is either time-independent or determined as solutions to inhomogeneous wave equations. In addition, this result is applied to nonlinear KGS system and the unique solvability in the energy space is proved.Then, we study the conserved KGS system in R3+1 in Chapter 4. For the per-turbed Schrodinger equation, we introduce the so called atomic spaces, construct a local Strichartz estimate on the dyadic block and obtain the low regular prop-erties for the energy solutions. By using dyadic multipliers and Littlewood-Paley decomposition in frequency, we obtain the regular estimates in atomic spaces for the nonlinear terms. Finally, we consider the KGS system in dinucleon field, combine compactness discussion with the Lipschitz estimate in a weak topology to prove the uniqueness and continuous dependence on the initial data.Finally, the higher order Yukawa KGS-type system is discussed in Chapter 5. By establishing a modified Virial type equation, the blow-up alternative for the radial solutions in the energy space is proved. Furthermore, on the assumption that blow-up occurs on the finite time, the blow-up rate of the regular solutions in three-dimensional space is obtained.