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.
Categories: math.AP