The concentration $c(x, t)$ of a chemical in one dimension obeys the equations

$$\frac{\partial c}{\partial t}=\frac{\partial}{\partial x}\left(c^{2} \frac{\partial c}{\partial x}\right), \quad \int_{-\infty}^{\infty} c(x, t) d x=1$$

State the physical interpretation of each equation.

Seek a similarity solution of the form $c=t^{\alpha} f(\xi)$, where $\xi=t^{\beta} x$. Find equations involving $\alpha$ and $\beta$ from the differential equation and the integral. Show that these are satisfied by $\alpha=\beta=-1 / 4$.

Find the solution for $f(\xi)$. Find and sketch the solution for $c(x, t)$.