Global behavior of solutions to the focusing generalized Hartree equation

We study the global behavior of solutions to the nonlinear generalized Hartree equation, where the nonlinearity is of the non-local type and is expressed as a convolution, $$i u_t + \Delta u + (|x|^{-(N-\gamma)} \ast |u|^p)|u|^{p-2}u=0, \quad x \in \mathbb{R}^N, t\in \mathbb{R}.$$ Our main goal is to understand behavior of $H^1$ (finite energy) solutions of this equation in various settings. In this work we make an initial attempt towards this goal. We first investigate the $H^1$ local wellposedness and small data theory. We then, in the intercritical regime ($0<s<1$), classify the behavior of $H^1$ solutions under the mass-energy assumption $\mathcal{ME}[u_0]<1$, identifying the sharp threshold for global versus finite time solutions via the sharp constant of the corresponding convolution type Gagliardo-Nirenberg interpolation inequality (note that the uniqueness of a ground state is not known in the general case). In particular, depending on the size of the initial mass and gradient, solutions will either exist for all time and scatter in $H^1$, or blow up in finite time or diverge along an infinity time sequence. To either obtain $H^1$ scattering or divergence to infinity, in this paper we employ the well-known concentration compactness and rigidity method of Kenig-Merle [36] with the novelty of studying the nonlocal nonlinear potential given via convolution with negative powers of $|x|$ and different, including fractional, powers of nonlinearities.