On the semirelativistic Hartree-type equation

We study the global Cauchy problem and scattering problem for the semirelativistic Hartree-type equation in R-n, n = 1, with nonlocal nonlinearity F(u) = lambda(vertical bar x vertical bar(-gamma) * vertical bar u vertical bar(2)) u, 0 < gamma < n. We prove the existence and uniqueness of global solutions for 0 < gamma < (2n)/(n + 1), n >= 2 or gamma > 2, n >= 3, and the nonexistence of asymptotically-free solutions for 0 < gamma <= 1, n >= 3. We also specify asymptotic behavior of solutions as the mass tends to zero and in. nity.