Consider some value $p=p(x,t)$ that follows the following PIDE ( $t$ representing time, the other one being spatial):
$$\frac{\partial p}{\partial t} = \frac{\partial}{\partial x} (xp(x,t)) + \int_{0}^{x}p(t,y) \ dy$$
It seems pretty clear to tme that, to determine the temporal dependence (at least, one possible one) of $p$, you can simply choose $p(x,t) = p(x)e^{A(t-t_0)}$ for some undetermined $p(x)$ such that it follows:
$$Ap(x) = \frac{\partial}{\partial x} (xp(x)) + \int_{0}^{x}p(y) \ dy $$
Am I missing anything here?