Well-posedness And Asymptotic Behavior For Some Nonlinear Evolution Equations

Evolution equations describe the states or processes in physics and other scientific fields that evolve as the time goes, and are a collective name of many important par-tial differential equations that depend on the time variable. Lots of partial differential equations that model complex phenomena are nonlinear. Nonlinear partial differential equations are the frontier and hot topic of nonlinear sciences. The KdV equation, KdV-BO equation and DS equation are important nonlinear evolution equations. They all are of important significances both in the descriptions of physical phenomena and in the investigations for mathematical theories. In the past few decades, more and more investi-gations are devoted to these equations, which promotes the fast developing of the related theories. In this dissertation, we mainly study the well-posedness of these equations and the time behavior of the solutions.In Chapter 2, we consider the local well-posedness and global well-posedness for the Cauchy problem for a higher order shallow water type equation on circleThe equation is a higher order modification of the Camassa-Holm equation. First, we establish some bilinear estimates for the nonlinear terms, then use these estimates to prove the Cauchy problem is locally well-posed in spaces Hs(T)(s≥-(j-2)/2) for arbitrary initial data. Next, making use of I-method, and by establishing the almost conservation law on the energy of the solution, we derive that the Cauchy problem to the equation is globally well-posed in spaceIn Chapter 3, we consider the local well-posedness for the Cauchy problem to the following stochastic KdV equation in Sobolev spaces Hs(R,)(s> -3/4). Under the frame of Bourgain’s spaces, and the assumption that the initial data belong to HS(H) almost everywhere, we prove by es-tablishing careful bilinear estimations the local well-posedness of the stochastic KdV equation in spaces. Due to the lack of smoothness in time for solutions of the linear stochastic KdV equation, the bilinear esti-mates for the nonlinear term 1/2(?)x(u2) become more difficult. By introducing spaces Xs,b and Xs,s,b(s< 0), we succeed in dealing with the low frequencies caused by the high-high frequency interaction arising inside the convolution integral in the expression of the Fourier transform of 1/2(?)x(u2), and eventually we overcome the difficulty and obtain the conclusion in this chapter.In Chapter 4, we consider the local well-posedness for the Cauchy problem to the ollowing stochastic KdV-BO equationin Sobolev spaces Hs(R)(s>-3/4). Although this equation is more complicated than the equation studied in Chapter 3, still we can find some similarities in form between them. Actually, we mainly adopt the method in Chapter 3 to study the above equation. The main difficulty comes from the bilinear estimate for the nonlinear term 1/2(?)x(u2), and we use a method analogous to that of Chapter 3 to conquer it. For the terms appear in the equation only, we use some additional tools to deal with the estimates related to them.In Chapter 5, we consider the minimal mass blow-up solution to the elliptic-elliptic Davey-Stewartson system in 2 space dimensions By the ground states correspond to the above system, we give out the specific forms of the minimal mass blow-up solutions at the end. In this process, the main difficulty comes from the fact that the uniqueness problem for the ground states correspond to ;he Davey-Stewartson system is still unsolved. In order to overcome the difficulty, we ise the principle of concentration compactness and the variational characterization of :he ground states correspond to the Davey-Stewartson system. Then combining some technical inequalities, we prove that each of the blow-up solutions has only one blow-up point, which helps us to overcome the difficulty and get the conclusion in this chapter.In Chapter 6, we study the problem of threshold of global existence and blow-up for the following generalization of Davey-Stewartson systemwhere t≥0, x= (x1, X2,x3)∈R3, and 1<p<7/3. Finally, we obtain some clear criteria, such that when the initial data of the system satisfy the conditions in the criteria, the corresponding solution exists globally or blows up at a finite time. We finish it in two steps. First, by analyzing the properties of the upper bound function of the energy, we construct two kinds of invariant sets for the generalized Davey-Stewartson system; Second, we show that when the initial data to the system belong to one kind of the invariant sets, the corresponding solution exists globally, while blows up in finite time when the initial data belong to the other. An careful discussion on the relationship between the upper bound function of the energy E(u(t)) and the upper bound function of V"(t), is critical in obtaining the conclusions of this chapter.