I would like to solve the PDE
$$ \frac{\partial y(x, t)}{\partial t} = \int dx' \lambda(x - x') \rho(x') y(x', t) $$
subject to some initial condition at $t=0$, where $\lambda$ is a kernel that specified how different parts of the space affect one another, $\rho$ is a weight function, and $y$ is some field whose evolution I am interested in. The coordinates $x$ above are vectors in $\mathbb{R}^q$.
The convolution of $\lambda$ with $\rho y$ suggests that taking a Fourier transform could help to solve the problem. However, in Fourier space, the product of $\rho$ and $y$ ends up becoming a convolution.
Any ideas on how to make progress with this problem?