= Solution
<Separation of variables> gives $X''=-k^2X$ and $T'=-\kappa k^2T$. The periodic matching of both the <function> and its first <derivative> selects $k=n\pi/L$, with the constant mode at $n=0$. Thus the <periodic diffusion on a circle> solution is
$$
u(x,t)=\frac{a_0}{2}+\sum_{n=1}^{\infty}\left[a_n\cos\frac{n\pi x}{L}+b_n\sin\frac{n\pi x}{L}\right]e^{-\kappa(n\pi/L)^2t},
$$
where the <Fourier coefficients> of the initial <function> are
$$
a_n=\frac1L\int_{-L}^Lu_0(x)\cos\frac{n\pi x}{L}\,dx,\qquad b_n=\frac1L\int_{-L}^Lu_0(x)\sin\frac{n\pi x}{L}\,dx.
$$
The formula for $a_n$ also gives $a_0$ at $n=0$. The constant mode preserves the initial spatial <mean>; every nonconstant <Fourier mode> decays. For initial data in <L2 space>, the series approaches those data in <L2 space> as $t\downarrow0$ and is smooth at positive times.
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