I am wondering if there are good methods (analytical, numerical) to solve equations of the form
$$ f(\mathbf{x})\frac{\partial u(\mathbf{x},t)}{\partial t} + \mathbf{a}(\mathbf{x})\cdot \nabla u(\mathbf{x},t) + b(\mathbf{x},t)u(\mathbf{x},t) + \iiint u(\mathbf{x}',t)\,d^3\mathbf{x}' = c_0 ,\quad u(\mathbf{x},t)=u_0(\mathbf{x}) ,\quad \mathbf{x}\in\mathbb{R}^3 ,\quad t\in[0,\infty). $$
In the case where the integral is constant in time (thus a PDE, no more IPDE), I already found the solution using the method of characteristics. Having the integral present makes things difficult, and I am not sure whether my previous method can be extended. Good numerical methods would also be great whether or not the analytic solution can be written.