A Note On The Inhomogeneous Hartree Equations With Potential

We consider the focusing non-homogeneous Hartree equation with potential i∂tv −ℋVu = −|v|p−2 1 |x|η 𠒥α ∗|v|p 1 |⋅|η u,ℋV = −Δ + V.Here, the time-space variable is (t,x) ∈ ℠× ℠3, and the potential V satisfies certain conditions that ensure the linear Schrödinger operator ℋV has dispersive and Strichartz estimates. The exponent η > 0 introduces a decaying singular term. The source term is assumed to be inter-critical, i.e. 2−2η+α 3 < p − 1 < 2−2η+α 3. The Riesz potential is defined on ℠3 by 𠒥α(x) := Cα|x|α−3 for some 0 < α < 3.The objectives of this work are two-fold: first, to develop a local well-posedness theory in the energy space HV1 := {f ∈ L2(℠3) |ℋ Vf ∈ L2(℠3)}; 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 V.The challenges addressed here include dealing with the issues: a convolution nonlinear term, the singular term 1 |⋅|η that breaks the space translation invariance of the equation, and the presence of the potential V, 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 V ∈{0,|x|−2}.