Guixiang Xu

Global well-posedness and scattering for the mass-critical Hartree equation with radial data

Abstract We establish global well-posedness and scattering for solutions to the mass-critical nonlinear Hartree equation i u t + Δ u = ± ( | x | −2 ∗ | u | 2 ) u for large spherically symmetric L x 2 ( R d ) initial data; in the focusing case we require, of course, that the mass is strictly less than that of the ground state.

Global well-posedness and scattering for the focusing energy-critical nonlinear Schrödinger equations of fourth order in the radial case

We consider the focusing energy-critical nonlinear Schrodinger equation of fourth order iut+Δ2u=|u|8d−4u, d⩾5. We prove that if a maximal-lifespan radial solution u:I×Rd→C obeys supt∈I‖Δu(t)‖2<‖ΔW‖2, then it is global and scatters both forward and backward in time. Here W denotes the ground state, which is a stationary solution of the equation. In particular, if a solution has both energy and kinetic energy less than those of the ground state W at some point in time, then the solution is global and scatters.

Global well-posedness and scattering for the energy-critical, defocusing Hartree equation for radial data

Abstract We consider the defocusing, H ˙ 1 -critical Hartree equation for the radial data in all dimensions ( n ⩾ 5 ) . We show the global well-posedness and scattering results in the energy space. The new ingredient in this paper is that we first take advantage of the term − ∫ I ∫ | x | ⩽ A | I | 1 / 2 | u | 2 Δ ( 1 | x | ) d x d t in the localized Morawetz identity to rule out the possibility of energy concentration, instead of the classical Morawetz estimate dependent of the nonlinearity.

Global Well-Posedness and Scattering for the Energy-Critical, Defocusing Hartree Equation in ℝ1+n

Using the same induction on energy argument in both the frequency space and the spatial space simultaneously as in [6, 33, 38], we obtain the global well-posedness and scattering of energy solutions of the defocusing energy-critical nonlinear Hartree equation in ℝ × ℝ n (n ≥ 5), which removes the radial assumption on the data in [25]. The new ingredients are that we use a modified long time perturbation theory to obtain the frequency localization (Proposition 3.1 and Corollary 3.1) of the minimal energy blow up solutions, which cannot be obtained from the classical long time perturbation and bilinear estimate and that we obtain the spatial concentration of minimal energy blow up solution after proving that -norm of minimal energy blow up solutions is bounded from below, the -norm is stronger than the potential energy.

Global well-posedness, scattering and blow-up for the energy-critical, focusing Hartree equation in the radial case

We establish global existence, scattering for radial solutions to the energy-critical focusing Hartree equation with energy and

Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equations of fourth order in dimensions d⩾9

\dot{H}^1

On the blow-up phenomenon for the mass-critical focusing Hartree equation in ℝ⁴

norm less than those of the ground state in

Global well-posedness and scattering for the defocusing H12-subcritical Hartree equation in Rd

\mathbb{R}\times \mathbb{R}^d

The Dynamics of the 3D Radial NLS with the Combined Terms

,

The dynamics of the NLS with the combined terms in five and higher dimensions

d\geq 5