This problem concerns heat conduction in a medium where the heat conduction coefficient depends algebraically on temperature: $\kappa=\kappa(T)=A T^{n}$. (For Maxwell molecules, hard spheres, and very hard particles, the exponent is $n=$ $1,1 / 2,0$, respectively, following the argument that led to (3.57) for the diffusion
coefficient.) The heat conduction equation
$$
\frac{\partial T}{\partial t}=A \nabla \cdot\left(T^{n} \nabla T\right)
$$
is mathematically a parabolic partial differential equation. When $n=0$, there is a general solution for an arbitrary initial condition. When $n \neq 0$, the resulting nonlinear parabolic equation is generally insoluble, but exact solutions are known for sufficiently simple initial conditions. Consider one such situation where heat is instantaneously released on the (hyper)plane $x=0$ while outside the initial temperature is zero. Then we need to solve the initial value problem
$$
\frac{\partial T}{\partial t}=A \frac{\partial}{\partial x}\left(T^{n} \frac{\partial T}{\partial x}\right), \quad T(x, 0)=Q \delta(x)
$$.
The goal is to solve this initial value problem when $n \geq 0$.
(a) On general grounds, $T=T(x, t \mid A, Q)$. Using dimensional analysis show that any three out of four quantities $x, t, A, Q$ have independent dimensions.
(b) Choosing $t, A, Q$ as the basic quantities, show that the solution must have the form
$$
T(x, t)=\left(\frac{Q^{2}}{A t}\right)^{1 /(2+n)} f(\xi), \quad \xi=\frac{x}{\left(Q^{n} A t\right)^{1 /(2+n)}}
$$
Thus dimensional analysis alone shows that the solution (which in principle depends on two variables $x$ and $t$ ) is, up to the scaling factor, a function of a single scaling variable $\xi$. The primary reason for this simplification is the absence of a quantity with dimensions of length in the formulation of the initial value problem.
(c) Substitute the above ansatz for $T(x, t)$ into the nonlinear heat equation and show that the scaling function satisfies the ordinary differential equation
$$
\left(f^{n} f^{\prime}\right)^{\prime}+\frac{1}{2+n}(\xi f)^{\prime}=0
$$
where the prime denotes differentiation with respect to $\xi$.
(d) Show that the solution to the above differential equation is
$$
f(\xi)=\left[\frac{n}{2(2+n)}\left(\xi_{0}^{2}-\xi^{2}\right)\right]^{1 / n}
$$
in the region $|\xi| \leq \xi_{0}$ and $f(\xi)=0$ for $|\xi| \geq \xi_{0}$. Argue that $\xi_{0}$ is determined by heat conservation.
$$
Q=\int T(x, t) d x=Q \int_{-\xi_{0}}^{\xi_{0}} f(\xi) d \xi
$$
and compute $\xi_{0}$ -
(e) The heated region is $\left[-x_{0}(t), x_{0}(t)\right]$ whose boundaries grow as $x_{0}(t)=$ $\xi_{0}\left(Q^{n} A t\right)^{1 /(2+n)}$; outside the heated region the medium remains at zero temperature. Try to explain the contradiction with the common wisdom that, for a parabolic partial differential equation, perturbations propagate instantaneously.