 Gradient flow of the periodic screened Poisson energy

For smooth $2\pi$-periodic $f$, the equation $u_t-u_{xx}+u=f$ is the negative $L^2$ gradient flow of $I_f[u]=\frac{1}{2}\int (u_x^2+u^2-2fu)dx$. Periodic integration by parts gives $d{I}_f/dt=-\int u_t^{2}dx$. Its unique equilibrium has Fourier coefficients $f_n/(1+n^2)$, and every transient coefficient decays as $e^{-(1+n^2)t}$. Thus the flow converges to the unique minimizer of the energy.