# Remarks on the relativistic Hartree equations

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REMARKS ON THE RELATIVISTIC HARTREE EQUATIONS YONGGEUN CHO, TOHRU OZAWA, HIRONOBU SASAKI, AND YONG-SUN SHIM Abstract. We study the global well-posedness (GWP) and small data scat- tering of radial solutions of the relativistic Hartree type equations with nonlo- cal nonlinearity F (u) = λ(| · |−γ ∗ |u|2)u, λ ∈ R \ {0}, 0 < γ < n, n ≥ 3. We establish a weighted L2 Strichartz estimate applicable to non-radial functions and some fractional integral estimates for radial functions. 1. Introduction In this paper, we consider the Cauchy problems concerning the relativistic Hartree equations: { i∂tu = √ 1−∆u+ F (u) in Rn × R, n ≥ 3, u(0) = ϕ, (1) { ∂2t u+ (1−∆)u = F (u) in Rn × R, n ≥ 3, u(0) = ϕ1, ∂tu(0) = ϕ2. (2) The nonlinear part F (u) is of Hartree type such that F (u) = Vγ(u)u, where Vγ(u)(x) = λ(| · |−γ ∗ |u|2)(x) = λ ∫ Rn |u(y)|2 |x− y|γ dy. Here λ is a non-zero real number and γ is a positive number less than the space dimension n. The first equation (1) is called the semi-relativistic equation which describes the Boson stars [6, 7, 13] and the second one (2) is the well-known Klein-Gordon equation whose physical model is originated from the helium atom [10, 14, 17]. For the simplicity of presentation, the mass, speed of light and Planck constant of both equations have been normalized. The equations (1) and (2) can be rewritten in the form of the integral equations u(t) = U(t)ϕ− i ∫ t 0 U(t− t′)F (u)(t′)dt′,(3) u(t) = (cos tω)ϕ1 + ω−1(sin tω)ϕ2 − ∫ t 0 ω−1(sin(t− t′)ω)F (u) dt′,(4) 1991 Mathematics Subject Classification. Primary: 35Q40, 35Q55; Secondary: 47J35. Key words and phrases. relativistic Hartree type equations, global well-posedness, scattering, radial solutions. 1 2 Y. CHO, T. OZAWA, H. SASAKI, AND Y.-S. SHIM where ω = √ 1−∆ and the associated unitary group U(t) is realized by the Fourier transform as U(t)ϕ = (e−itωϕ)(x) ≡ 1 (2pi)n ∫ Rn eix·ξe−it √ 1+|ξ|2 ϕ̂(ξ) dξ, where ĝ denotes the Fourier transform of g defined by ĝ(ξ) =

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