Small data scattering for semi-relativistic equations with Hartree type nonlinearity

Abstract

We prove that the initial value problem for the equation \[ - i\partial_t u + \sqrt{m^2-Δ} \, u= (\frac{e^{-μ_0 |x|}}{|x|} \ast |u|^2)u \ \text{in} \ \mathbb R^{1+3}, \quad m\ge 0, \ μ_0 >0\] is globally well-posed and the solution scatters to free waves asymptotically as $t\to \pm \infty$ if we start with initial data which is small in $H^s({\mathbb{R}}^3)$ for $s>\frac{1}{2}$, and if $m>0$. Moreover, if the initial data is radially symmetric we can improve the above result to $m\ge 0$ and $s>0$, which is almost optimal, in the sense that $L^2({\mathbb{R}}^3)$ is the critical space for the equation. The main ingredients in the proof are certain endpoint Strichartz estimates, $L^2({\mathbb{R}}^{1+3})$ bilinear estimates for free waves and the application of the $U^p$ and $V^p$ function spaces.

19 pages

References in corpus (4)

• The cubic Dirac equation: Small initial data in $H^{\frac{1}{2}}({\mathbb{R}}^2)$
• Effective Dynamics for Boson Stars
• Angular Regularity and Strichartz Estimates for the Wave Equation
• Global existence without decay for quadratic Klein-Gordon equations

Cited by in corpus (6)

• Small data scattering of 2d Hartree type Dirac equations
• On the modified scattering of $3$-d Hartree type fractional Schrödinger equations with Coulomb potential
• Critical well-posedness and scattering results for fractional Hartree-type equations
• The fractional in time Schrödinger equation with a Hartree perturbation
• Small data scattering of semirelativistic Hartree equation
• Scattering results for Dirac Hartree-type equations with small initial data