Partial Integro-Differential Equation

[NeoDevin]

I need a little help solving an equation here, I don't really know where to start. If anyone has any advice on solving (or even simplifying) such a beast, it would be much appreciated.

$$\frac{\partial}{\partial t}N + \frac{1}{\tau}N = -a(t)\frac{\partial}{\partial u}N - u\frac{\partial}{\partial x}N -v\frac{\partial}{\partial y}N + \frac{1}{\tau}D(u,v)\int_{-\infty}^\infty\int_{-\infty}^\infty Ndudv$$
Where
$$N = N(u,v,x,y,t)$$

$D(u,v)$ Is a given normalized distribution function. I get to choose this, if you need a particular example, use:
$$D(u,v) = \frac{1}{2\pi v_{th}^2}e^{-(u^2+v^2)/2v_{th}^2}$$

$$a(t) = A_0\cos(\omega t)$$

$A_0$, $v_{th}$, and $\tau$ are all positive real constants.

And the solution $N$ must be normalizable over $u$ and $v$

 

[NeoDevin]

What would be the boundary conditions to model a boundary with 100% specular reflection?