Solving for moments of $u(x,t)$ given the equation $u_t=xu+f\ast u$

Question

Given the evolution equation $$ \frac{\partial u}{\partial t}(x,t)=xu(x,t)+\int_{-\infty}^\infty f(x-y)u(y,t)dy $$ with initial condition $u(x,t)=\delta(x-x_0)$, is there a way to solve for either or both of the quantities $$ m_0(t)=\int_{-\infty}^\infty u(x,t)dx \quad\text{ or }\quad m_1(t)=\int_{-\infty}^\infty xu(x,t)dx \quad $$ in closed form, in terms of the positive-valued function $f(\cdot)$?