I want to solve below equation:
$$ \partial_t u(\vec{\rho},t)=\nabla^2u - \nabla \cdot (u U) $$
where
$$ U = \int_a^L \frac{u \vec{\rho}}{(\rho^2+a^2)^{3/2}} \rho d\rho d\phi$$
$\phi$ is angle between $\rho$ and $x$ axis. I tried to solve it using Laplace transform but as the problem is not linear in integral part, I failed to do that. How can I solve it? Is is a solveable or not? $\vec{\rho}$ is radial vector in polar coordinate.