This problem was asked in a test of the course I'm attending of PDE, but I could not solve it and I would like to know how to do it.
A semi infinite bar has an initial temperature of zero. At $t = 0$, the temperature is suddenly raised to $T_0$ at the boundary $x = 0$. This temperature is conserved until a time $t_0$ after which it reduces to zero. Prove that, for some additional time after $t_0$, the temperature has a maximum at a distance $x = \sqrt{kt_0\ln(2)}$, where $k$ is the diffusivity and is considered constant.
My attempt was to solve the PDE problem
\begin{align} u_t &= k^2u_{xx}, \quad 0 < x < \infty, t > 0 \\ u(0,t) &= T_0[ H(t) - H(t - t_{0}) ] \\ u(x,0) &= 0, \quad 0 < x < \infty \end{align}
where $H(t-t_{0})$ is the Heaviside function.
Following example 1, page 503 from the book 'Partial Differential Equations with Fourier Series and Boundary Value Problems' (it uses the Laplace transform and solves the ODE, and then uses the convolution theorem), the problem has the solution
\begin{align} u(x,t) &= \frac{x}{2k\sqrt{\pi}} \int_{0}^{t} \frac{T_0 \big[H(t) - H(t - t_{0})\big] \exp{ \left( \frac{-x^2}{4c^2(t - \tau) } \right) }}{(t - \tau)^{3/2}} d\tau \end{align}
I also tried to find the maximum of this function, but my approach didn't work. Thanks for the help.