Blow-up Solutions Of The Nonlinear Hartree Equation With A Perturbation

In this thesis,we consider the Cauchy problem for the nonlinear Hartree equation with a perturbation.We find the sharp energy threshold for the global existence and blowup to the nonlinear Hartree equation with a focusing and defocusing perturbation,respectively,by utilizing comprehensive potential well structures generated by the combined nonlinearities.Furthermore,we applied the refined compactness lemma to obtain the dynamical properties to the nonlinear Hartree equation with a perturbation,including blow-up rate,the weak limit and the limiting profile of blow-up solutions.The paper was divided into the following four chapters.In Chapter 1,we introduce the research background,main contents and main work.In Chapter 2,as preliminaries,we recall the Strichartz estimates,local well-posedness and the variational characteristic of the ground state solutions to the corresponding elliptic equation.In Chapter 3,by overcoming the difficulties of complex potential well structures derived the perturbation,we construct invariant sets of the Cauchy problem,and then obtain the sharp energy thresholds for the nonlinear Hartree equation in the case of different parameters.In Chapter 4,we use Strichartz estimates,variational characteristic of the ground state solutions and refined compactness lemma to obtain dynamical properties of blow-up solutions,including blow-up rates,weak limits and limiting profile of blow-up solutions with small super critical mass.