Boundary Values & Fourier Series · Superchallenge problem

A string $0 \le x \le \pi$ with fixed ends hangs flat and at rest until, at $t = 0$, a uniform load is switched on: $u_{tt} = u_{xx} + 1$, $u(0,t) = u(\pi,t) = 0$, $u(x,0) = u_t(x,0) = 0$. What is the largest displacement the midpoint ever reaches, $\displaystyle\max_{t\ge0}u\!\left(\dfrac\pi2, t\right)$?