Consider a nonlinear model for the axisymmetric dispersal of a population in two spatial dimensions whose density, $n(r, t)$, obeys

$$\frac{\partial n}{\partial t}=D \boldsymbol{\nabla} \cdot(n \boldsymbol{\nabla} n)$$

where $D$ is a positive constant, $r$ is a radial polar coordinate, and $t$ is time.

Show that

$$2 \pi \int_{0}^{\infty} n(r, t) r d r=N$$

is constant. Interpret this condition.

Show that a similarity solution of the form

$$n(r, t)=\left(\frac{N}{D t}\right)^{1 / 2} f\left(\frac{r}{(N D t)^{1 / 4}}\right)$$

is valid for $t>0$ provided that the scaling function $f(x)$ satisfies

$$\frac{d}{d x}\left(x f \frac{d f}{d x}+\frac{1}{4} x^{2} f\right)=0 .$$

Show that there exists a value $x_{0}$ (which need not be evaluated) such that $f(x)>0$ for $x<x_{0}$ but $f(x)=0$ for $x>x_{0}$. Determine the area within which $n(r, t)>0$ at time $t$ in terms of $x_{0}$.

[Hint: The gradient and divergence operators in cylindrical polar coordinates act on radial functions $f$ and $g$ as

$$\left.\boldsymbol{\nabla} f(r)=\frac{\partial f}{\partial r} \hat{\boldsymbol{r}} \quad, \quad \boldsymbol{\nabla} \cdot[g(r) \hat{\boldsymbol{r}}]=\frac{1}{r} \frac{\partial}{\partial r}(r g(r)) .\right]$$