Sharp Conditions For Global Existence Of Nonlinear Wave System

The nonlinear Schrodinger equation is a typical dispersive wave equation, which reflects the relation between dispersion and nonlinear interaction. At the same time, the nonlinear Schrodinger equation is also a fundamental model in quantum mechanics, and different nonlinear Schrodinger equations have different physical backgrounds. For instance, classical nonlinear Schrodinger equation is used to describe some phenomena in quantum physics, such as the propagation of laser beam in dispersive and nonlinear medium,self trapping in nonlinear optics( [44])and Langmur waves in plasma( [89]). The nonlinear Schrodinger equation with harmonic potential is known as a model for describing the remarkable Bose-Einstein condensate(BEC) ( [11,12,81,84,94]), The second order derivative nonlinear Schrodinger equations is used to describe the upper-hybrid oscillation propagation( [67,51,45]). The generalized Davey-Stewartson system describes the evolution of weakly nonlinear water waves that travel predominantly in one direction( [26,31]), the nonlinear Schrodinger equation with white noise models the propagation of nonlinear dispersive waves in nonhomogeneous or random media( [1]).In the recent thirty years, a series of important advances are achieved on the mathematical studies of the classical nonlinear Schrodinger equation. Especially for its typical properties such as the local well-posedness of the initial value problems, existence of the global solutions and their asymptotic behavior, existence of the standing waves and their stability, blow-up properties of the solutions in a finite time and their dynamical behavior, plentiful and substantial results are got. Ginibre and Velo [32] established the local existence in energy space H~1(R~N). Ginibre and Velo [32], Lin, Strauss [49], Strauss [76], Tsutsumi [82], Cazenave [22] discussed the asymptotic properties. Strauss [75], Berestycki, Lions [3,4] studied the existence of standing waves, and Kwong [46] obtained the uniqueness of the ground state solution; Berestycki, Cazenave [2], Cazenave, Lions [20], Weinstein [87], Shatah, Strauss [75], Grikllakis, Shatah, Strauss [35,36] studied the stabilities of standing waves. Glassey [34], Weinstein [85], Merle [53,54], Zhang [96,97], Ogawa, Tsutsumi [57,58] discussed the existence of blow-up solutions.For the nonlinear Schrodinger equation with harmonic potential, Oh [59] established local existence in the corresponding energy space. Rabinowitz [69], Cao, Noussair [13]studied the existence of standing waves. Rose, Weinstein [71], Zhang [94,95] studied the stability of standing waves. Tsutsumi, Wadati [81], Zhang [94,98], Carles [14,15,16,17], Cazenave [18] discussed the existence of blowup solutions.Chen,Zhang [23,24,25]got the sharp conditions of global existence.For the second order derivative nonlinear Schrodinger equation, most of papers are on the blow-up and well-posedness of solutions, for instance, Laedke and Spatschek [47] got the Liapunov stability of soliton solutions, Poppenberg [64,65] got the local and global well-posedness results concerning the Cauchy problem in H~∞and H~k(R~N), Zhang [92] got the blow-up properties of solutions in arbitrary dimension, and global existence of solutions in two or three dimensions is given in [30].For the generalized Davey-Stewartson system, Guo and Wang [37], Ghidaglia and Saut [31] established the local existence in energy space H~1(R~N). Ohta [61] proved that if p > 3, there exists blow-up solutions arbitrarily close to the standing wave in the case of N = 2. In addition, Ohta [62] also proved that if p≥1 + 4/N, the ground stateφ_ωis unstable for anyω∈(0.∞) in the case of N = 2 or N = 3.For the stochastic nonlinear Schrodinger equation with white noise, de Bouard and Debussche [9] established the local existence in energy space H~1 and global existence in subcritical or defocusing case. In addition, de Bouard and Debussche [7,8]also proved that for some initial data, blow-up occurs in the sense that, for arbitrary t > 0, the probability that the solution blows up before the time t is strictly positive.From the view-point of physics, for nonlinear wave system, the following problems are very important. Under what conditions, will the wave become unstable to collapse(blow up)? And under what conditions, will the wave exits for all time(global existence)? Especially the sharp criteria for blow-up and global existence are pursued strongly.In this paper, we shall study several types of known nonlinear Schrodinger equations. Our method and argument are originated in the framework of combining the global well-posedness of the nonlinear wave system with standing waves based on the modern variation method, which is established by Zhang [94,96,97,98]. In the framework, we develop them. First analyzing the characteristics of these equations, basing on the local well-posedness of the Cauchy problem, we set some intricate functionals and Nehari manifolds to pose constrained variational problem. Then combining the characteristics of these equations, the variational problems, we construct some evolution invariant flows of these equations, and finally we derive the blow-up and global existence of solutions, instablility of standing waves, sharp conditions of global existence. This paper are organized as follows:In chapter 1, we present the physical background and some known results on these equations. In addition, main results of this paper are also given. In chapter 2, we discussed the nonlinear Schrodinger equation with harmonic potential. By constructing a cross-constrained variational problem and so-called invariant manifolds of the evolution flow, we derive a sharp criterion for blow-up and global existence of the solutions.In chapter 3, we studied the second order derivative nonlinear Schrodinger equation. By establishing a variational problem, applying the potential well argument and the concavity method, we prove that there exists a sharp condition for global existence and blow-up of the solutions. In addition, the question that how small the initial data are, the global solutions exist is also answered.In chapter 4, We studied the nonlinear Schrodinger equation with different power nonlinearities. We first establish the existence of standing wave associated with the ground states by variational calculus. Then by the potential well argument and the concavity method, we get a sharp condition for blow-up and global existence to the solutions and answer such a problem: how small are the initial data, the global solutions exist? At last we also prove the instability of standing wave by combing those results.In chapter 5, we considered the damped Gross-Pitaevskii(GP) equation. We prove that there exists a threshold value for the damping parameter and when the damping parameter is above the threshold value, the solutions of the Cauchy problem globally exist;when the damping parameter is under the threshold value,the solutions will blow up in a finite time.In chapter 6, we studied the generalized Davey-Stewartson system. By constructing a cross-constrained variational problem and so-called invariant manifolds of the evolution flow, we derive a sharp criterion for blow-up and global existence of the solutions.In chapter 7, we considered the coupled nonlinear Schrodinger equations. By constructing a cross-constrained variational problem and so-called invariant manifolds, we derive a sharp criterion for blow-up and global existence of the solutions. At the same time, the instability of standing wave is also given.In chapter 8, we studied the nonhomogeneous nonlinear Schrodinger equation. By constructing a constrained variational problem and so-called invariant manifolds of the evolution flow, we derive a sharp criterion for blow-up and global existence of the solutions.In chapter 9, we discussed the stochastic nonlinear Schrodinger equation with white noise. By the tools of stochastic analysis and the Gagliardo-Nirenberg inequality, we derive a sufficient condition for global existence of solutions .