The Nonlinear Schr(?)dinger Equations With Energy-critical Term

The nonlinear Schr¨odinger equation is the basic mathematic model in thequantum mechanics and also is one of the most active research topics in mathe-matical physics (see [2, 26, 66] and so on). The nonlinear Schro¨dinger equationis a typical dispersive wave equation, which re?ects the relation between dis-persion and nonlinear interaction. At the same time, as an important partialdi?erential equation, it is one of the most important parts of core mathematics.In the recent thirty years, the studying of nonlinear Schro¨dinger equationenter a new stage. From Segal pointed out the nonlinear semi-group theory[50], a series of important results have been got in the research of the energysubcritical Schr¨odinger equation. Ginbre and Velo established the local well-posedness[21, 22] in energy space. Subsequently, based on harmonic analysis,a series deep results[8, 13, 60] in the research of the existence of the global so-lutions and the asymptotic behaviour especially for scattering. For the blowupsolution of the energy subcritical nonlinear Schr¨odinger equation, there havegigantic advance: Glassey derived the su?cient condition[25], Ogawa and Tsut-sumi made the significantly improved[43, 44], and Merle and Raphael obtaineda series of important results on the dynamical properties[41, 42]. For the exis-tence of the standing wave and their stability, we can see the results obtained byCazenave and Lions in [15] and the symbolic results by Struass and other schol-ars in [27, 28]. Also, for the sharp condition of blowup and global existence ofthe energy subcritical nonlinear Schro¨dinger equations, there are plentiful and substantial results[65, 69–71], which have close relationship with the standingwave.For the energy-critical nonlinear Schr¨odinger equation, in 1990, Cazenaveand Weissler[14] developed the theory of the Cauchy problem for this equation.We can see that the energy-critical nonlinear Schr¨odinger equation is quite dif-ferent from the energy subcritical nonlinear Schr¨odinger equation, which bringsome new di?culties in our researches. When the energy-critical term is de-focussing, a series of achievements were made in the global existence and thescattering, see Bourgain[7], Tao[19, 59] and so on. When the energy-criticalterm is focussing, Kenig and Merle[32] studied the global existence, the scat-tering as well as the blowup properties; Duyckaerts and Merle[20] showed adynamical characterization of the explicit stationary solution.For the nonlinear Schr¨odinger equation with energy subcritical term andenergy-critical term, we called it nonlinear Schr¨odinger equation with energy-critical term. In three space, X.Zhang[72] studied the global existence, thescattering and the blowup properties. Tao, Visan and X.Zhang[62] discussed thelocal and global well-posedness, finite time blowup, and asymptotic behaviour ofthis Cauchy problem. Clearly, for the energy-critical term, the time of existencefor these local solutions of the nonlinear Schr¨odinger equation with energy-critical term depends on the profile of the data, rather than on its H1-norm; forthe energy subcritical term, this problem fail to be scale invariant. At the meantime, we find that when the energy-critical term is focussing, the solutions of thisequation globally exist for some initial data especially for a class of su?cientlysmall data, but the solutions of this equation blow up in a finite time for someinitial data especially for a class of su?ciently large data. Thus, in this paper,we discuss the sharp condition of blowup and global existence of the nonlinearSchr¨odinger equation with energy-critical term.As we all known, for the energy subcritical nonlinear Schr¨odinger equations,J. Zhang established a mature framework based on the variational method. Anda series of results have been obtained[17, 68–71]. But this framework can notsolve the problems of the nonlinear Schro¨dinger equation with energy-critical term completely. The variational method is not entirely e?ective, specially tothe proof of the global existence. Thus we use the Strichartz estimation skill.Then we try to approximate the nonlinear Schro¨dinger equation with energy-critical term by the energy-critical nonlinear Schr¨odinger equation with thesame initial data and then prove the global existence. More precisely, applyingthe long time stability(see in [61]), we treat the energy subcritical nonlinearity|u|p?1u as a perturbation to the energy-critical nonlinear Schr¨odinger equa-tion and get the global well-posedness. As for the blowup, by the frameworkwhich established by J.Zhang, combining some skills in [43, 44], we get theblowup results. Thus, in this paper, our method and argument are originatedin the framework of the constrained variational problem, which is establishedby J.Zhang, and combining the Strichartz estimation skill in harmonic analysis.By this method and argument we obtain the sharp condition of blowup andglobal existence for the Cauchy problem of the nonlinear Schr¨odinger equationwith energy-critical term when the initial data are subcritical energy and criticalenergy. This paper are organized as follows:In Chapter 1, we present the physical background and some known re-searching results of the nonlinear Schr¨odinger equation with a harmonic poten-tial. At the same time, we also present our goal and the main results of thepaper.In Chapter 2, we consider a energy-critical Schro¨dinger equations withenergy subcritical perturbations and address question related to the sharp cri-terion of global existence. By analyzing the variational characteristics of thisequation, we established a type of invariant ?ow. Then approximating theCauchy problem to the energy-critical nonlinear Schro¨dinger equations withthe same initial data and combining the properties of the invariant ?ows, weobtain the sharp conditions of global existence for the Cauchy problem whenthe initial data are subcritical energy. Moreover, we give a su?cient conditionto answer: How small is the initial datum such that the solution of the Cauchyproblem exists globally in time?In Chapter 3, we consider a energy-critical Schro¨dinger equations with energy subcritical perturbations and address question related to the sharp cri-terion of global existence. By analyzing the variational characteristics of thisequation, we established a type of invariant ?ow. And then we get the su?-cient conditions of the blowup results. Then approximating this equation tothe energy-critical nonlinear Schr¨odinger equations with the same initial data,we obtain the global existence for the Cauchy problem. By analyzing the re-sults above, we get the sharp conditions of blowup and global existence for theCauchy problem when the initial data are critical energy.In Chapter 4, we apply variational method to study the energy-criticalSchr¨odinger equations with energy subcritical perturbations. Through analyz-ing the Hamiltonian property we establish a type of invariant evolution ?ow,and derive a su?cient condition of blowup.In Chapter 5, we concerned with the Hartree type equations with energysubcritical perturbations. We propose a cross-invariant ?ow to study the sharpcriterion for global existence and strong instability of the standing waves tothe Hartree type equations.