Boundary Values & Fourier Series · Challenge problem

A rod $0 \le x \le \pi$ with ends held at $0$ starts cold, $u(x, 0) = 0$, and is heated by a fading source: $u_t = u_{xx} + e^{-t}\sin 2x$. Find the highest temperature ever reached at $x = \dfrac\pi4$, that is, $\displaystyle\max_{t\ge0}u\!\left(\dfrac\pi4, t\right)$.