Solution
= Solution
Differentiate under the integral and use the evolution equation:
$$
\frac d{dt}I_f[u(t)]=\int_{-\pi}^{\pi}(-u_{xx}+u-f)u_t\,dx=-\int_{-\pi}^{\pi}u_t^2\,dx\leq0.
$$
Thus $\boxed{I_f[u(t)]\leq I_f[u(s)]\ (t>s>0)}$. This is a <gradient flow of the periodic screened Poisson energy>. Part (iii) gives convergence in the function and its first derivative, so $\boxed{I_f[u(t)]\to I_f[u_f]}$, the minimum from part (ii).