A NOTE ON THE INHOMOGENEOUS HARTREE EQUATIONS WITH POTENTIAL

We consider the focusing non-homogeneous Hartree equation with potential [Formula: see text] Here, the time-space variable is [Formula: see text], and the potential [Formula: see text] satisfies certain conditions that ensure the linear Schrödinger operator [Formula: see text] has dispersive and Strichartz estimates. The exponent [Formula: see text] introduces a decaying singular term. The source term is assumed to be inter-critical, i.e. [Formula: see text]. The Riesz potential is defined on [Formula: see text] by [Formula: see text] for some [Formula: see text]. [Formula: see text]The objectives of this work are two-fold: first, to develop a local well-posedness theory in the energy space [Formula: see text]; and second, to present a dichotomy of global existence and scattering versus blow-up of solutions. Scattering is established using the new approach by Dodson–Murphy. The novelty of this work lies in the inclusion of the potential [Formula: see text]. [Formula: see text]The challenges addressed here include dealing with the issues: a convolution nonlinear term, the singular term [Formula: see text] that breaks the space translation invariance of the equation, and the presence of the potential [Formula: see text], which leads to a lack of scale invariance in the equation. This study naturally extends previous work by the authors on the same problem for [Formula: see text].