I have a function $v=v(x,y,t)$ satisfying the following PDE (degerate heat equation)
$\frac{\partial v}{\partial t} - \Delta_x v - \Delta_y v - 2 \nabla_x \cdot \nabla_y v = 0$.
I also have a function $r:[0,+\infty) \to [0,+\infty)$ such that $r(|x-y|)$ is of class $C^2$. The paper I am reading claims that using the PDE above and integration by parts, it is true that
$ \frac{d}{d t} \iint r(|x-y|) v(x,y,t) dx dy $
$= \iint v(x,y,t) \left(\Delta_x \left[ r (|x-y|) \right] + \Delta_y \left[ r (|x-y|) \right] + 2 \nabla_x \nabla_y [r(|x-y|)] \right) dx dy $
How can one show this?