A note on evolution equations with modified Hartree Nonlinearity

Abstract

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.