According to this paper, equation 16 says the 1 dimensional viscous wave equation is:
$\frac{4v}{3c_0^2}p_{xxt}+p_{xx}-\frac{1}{c_0^2}p_{tt}=0$
and equation 17 says the 3 dimensional viscous wave equation is:
$\frac{4v}{3c_0^2} \nabla^2 p_{t}+ \nabla^2 p-\frac{1}{c_0^2}p_{tt}=0$
But what does the 2 dimensional viscous wave equation look like?
I would guess that in every dimension $1,2,3,\dots$, the answer is just $$\frac{4v}{3c_0^2} \nabla^2 p_{t}+ \nabla^2 p-\frac{1}{c_0^2}p_{tt}=0$$ where the Laplacian is interpreted correctly for the dimension; $\frac{d^2}{dx^2}$ in dimension 1, $\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}$ in dimension 2, $\frac{\partial^2}{\partial x_1^2}+\frac{\partial^2}{\partial x_2^2}+\frac{\partial^2}{\partial x_3^2}$ in dimension 3, and so on.