Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 3 / 6C / Solution

For a radial function, the equation isTherefore, assuming the population density and radial flux decay at infinity and are regular at the origin,Thus $N$ is conserved; it is the total population obtained by integrating density over the plane. This is the global form of the equation's conservation law.

SetThenwhile the radial diffusion term has the same prefactor and equalsEquating them and multiplying by $x$ givesso the proposed ansatz is a similarity solution.

Regularity at $x=0$ makes the constant of integration zero. Wherever $f>0$,for some $x_0>0$. The nonnegative weak solution continues as $f=0$ once this parabola reaches zero:This is the two-dimensional Barenblatt profile for quadratic porous-medium diffusion; its degenerate diffusivity permits compact support.

Indeed, conservation requiresso $x_0=(16/\pi {)}^{1/4}$ if desired. At time $t$ the populated disk has radius $x_0(NDt{)}^{1/4}$ and hence area