Two-dimensional Barenblatt profile for quadratic porous-medium diffusion
= Two-dimensional Barenblatt profile for quadratic porous-medium diffusion
For the radially symmetric equation
$$
n_t=D\nabla\cdot(n\nabla n)
$$
in two dimensions, the mass-$N$ similarity solution has the compactly supported form
$$
n(r,t)=\left(\frac N{Dt}\right)^{1/2}
\left[\frac{x_0^2-x^2}{8}\right]_+,
\qquad
x=\frac r{(NDt)^{1/4}},
$$
where mass normalization gives $x_0=(16/\pi)^{1/4}$. Its support radius grows as $t^{1/4}$ and its occupied area grows as $t^{1/2}$.