The concentration $c(x, t)$ of insects at position $x$ at time $t$ satisfies the nonlinear diffusion equation

$$\frac{\partial c}{\partial t}=\frac{\partial}{\partial x}\left(c^{m} \frac{\partial c}{\partial x}\right)$$

with $m>0$. Find the value of $\alpha$ which allows a similarity solution of the form $c(x, t)=t^{\alpha} f(\xi)$, with $\xi=t^{\alpha} x$.

Show that

$$f(\xi)=\left\{\begin{array}{cl} {\left[\frac{\alpha m}{2}\left(\xi^{2}-\xi_{0}^{2}\right)\right]^{1 / m}} & \text { for }-\xi_{0}<\xi<\xi_{0} \\ 0 & \text { otherwise } \end{array}\right.$$

where $\xi_{0}$ is a constant. From the original partial differential equation, show that the total number of insects $c_{0}$ does not change in time. From this result, find a general expression relating $\xi_{0}$ and $c_{0}$. Find a closed-form solution for $\xi_{0}$ in the case $m=2$.