Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 3 / 29C / iii / Solution

The Fourier coefficients obey ${\dot{u}}_n+(1+n^2)u_n=f_n$, soThe coefficient decay of $u_0,f$ and the additional exponential damping justify all derivatives for $t>0$ and recovery of the initial data. Thus this is a smooth periodic solution. Every transient mode decays, giving $\boxed{u(t,\cdot )\to u_f}$ uniformly, indeed with all spatial derivatives. The elliptic solution in part (i) is the attracting steady state of this forced heat equation.