= Solution
The <Fourier coefficients> obey $\dot u_n+(1+n^2)u_n=f_n$, so
$$
\boxed{u(t,x)=\sum_{n\in\mathbb Z}\left[\frac{f_n}{1+n^2}+\left((u_0)_n-\frac{f_n}{1+n^2}\right)e^{-(1+n^2)t}\right]e^{inx}.}
$$
The 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>.
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