  We introduce a mathematical model in $\mathbb{R}^{n}$ for evolution equations
with modified generalized Hartree nonlinearity given by
$S_{\alpha,p,q}(u)=I_{\alpha}(|u|^{p+q}).$ One can see that this nonlinearity
is not integrable due to the boundedness property of Riesz potential. In other
words, we cannot deal with the Cauchy problem of semi-linear evolution
equations with $S_{\alpha,p,q}(u)$ and $L^{1}$-initial velocity.
  We will show that $S_{\alpha,p,q}(u)$ produces the same semi-critical
exponent that guarantees the global existence of small data solutions as in the
well known generalized Hartree nonlinearity
$H_{\alpha,p,q}(u)=|u|^{p}I_{\alpha}(|u|^{q})$ provided that the initial
velocity belongs to $L^{m}(\mathbb{R}^{n})$, with $m>1$.
  We can expect a relation between some physical systems that are modeled and
solved using Hartree nonlinearity and those in their modified form due to this
coincidence property in the semi-critical exponent.
