Sharp Threshold of Blow-up and Scattering for the fractional Hartree equation

Abstract

We consider the fractional Hartree equation in the $L^2$-supercritical case, and we find a sharp threshold of the scattering versus blow-up dichotomy for radial data: If $M[u_0{]}^{\frac{s-s_c}{s_c}}E[u_0<M[Q{]}^{\frac{s-s_c}{s_c}}E[Q]$ and $M[u_0{]}^{\frac{s-s_c}{s_c}}\Vert u_0{\Vert }_{{\dot{H}}^s}^2<M[Q{]}^{\frac{s-s_c}{s_c}}\Vert Q{\Vert }_{{\dot{H}}^s}^2$, then the solution $u(t)$ is globally well-posed and scatters; if $M[u_0{]}^{\frac{s-s_c}{s_c}}E[u_0]<M[Q{]}^{\frac{s-s_c}{s_c}}E[Q]$ and $M[u_0{]}^{\frac{s-s_c}{s_c}}\Vert u_0{\Vert }_{{\dot{H}}^s}^2>M[Q{]}^{\frac{s-s_c}{s_c}}\Vert Q{\Vert }_{{\dot{H}}^s}^2$, the solution $u(t)$ blows up in finite time. This condition is sharp in the sense that the solitary wave solution $e^{it}Q(x)$ is global but not scattering, which satisfies the equality in the above conditions. Here, $Q$ is the ground-state solution for the fractional Hartree equation.

Proposition 2.6 has been update

References in corpus (8)

• Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrodinger equation in the radial case
• A sharp condition for scattering of the radial 3d cubic nonlinear Schroedinger equation
• Decay estimates for a class of wave equations
• Global well-posedness and scattering for the energy-critical, defocusing Hartree equation for radial data
• Global well-posedness and scattering for the mass-critical Hartree equation with radial data
• Global well-posedness, scattering and blow-up for the energy-critical, focusing Hartree equation in the radial case
• Global well-posedness and scattering for the defocusing $H^{\frac{1}{2}}$-subcritical Hartree equation in ${\mathbb{R}}^d$
• Nonlinear fractional Schrödinger equations in one dimension

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• Orbital Stability of Standing Waves for a fourth-order nonlinear Schrödinger equation with the mixed dispersions
• Normalized ground states for the fractional nonlinear Schrödinger equations
• Strichartz estimates for orthonormal families of initial data and weighted oscillatory integral estimates
• On the blow-up solutions for the nonlinear Schrödinger equation with combined power-type nonlinearities
• Orbital Stability of Standing Waves for Fractional Hartree Equation with Unbounded Potentials
• Remarks on the fractional inhomogeneous Hartree equation