Abstract
This paper investigates the perturbed Hartree equation
i
u
˙
+
Δ
u
±
|
u
|
2
(
p
−
1
)
u
±
(
I
α
∗
|
u
|
q
)
|
u
|
q
−
2
u
=
0
.
Indeed, one addresses the questions of global well-posedness and scattering versus finite time blow-up of energy solutions. One needs to deal with the absence of scaling invariance and try to understand the concurrency between a local and non-local source terms. The global existence and scattering of energy critical solutions for small data are proved regardless the sign of the source terms. In the case of two attractive non-linearity, the scattering of global solutions is proved in the inter-critical regime. Moreover, a decay result in available in the mass-sub-critical case. When there is an attractive part and a repulsive part in the source term, one gives a dichotomy of global existence versus finite time blow-up of solutions and the strong instability of standing waves, depending on a comparison between the exponents
p
and
q
.