= Solution
For smooth periodic initial data, Fourier expansion reduces the equation to $\dot{\widehat u}(m,t)=-(1+m^2)\widehat u(m,t)$. Therefore define the <Fourier multiplier> solution operators by
$$
\widehat{S(t)u_0}(m)=e^{-(1+m^2)t}\widehat u_0(m).
$$
Their <Sobolev norm> satisfies $\|S(t)u_0\|_{H^s}^2\leq e^{-2t}\|u_0\|_{H^s}^2\leq\|u_0\|_{H^s}^2$, so they extend to <contractive linear operators> on $H^s_{\rm per}$. Multiplication of the Fourier factors gives $S(t+s)=S(t)S(s)$ and $S(0)=I$. Finally
$$
\|S(t)u-u\|_{H^s}^2=\sum_m(1+m^2)^s|e^{-(1+m^2)t}-1|^2|\widehat u(m)|^2\to0
$$
by <dominated convergence theorem>, dominated by four times the summable norm terms. This proves a \b[strongly continuous contraction semigroup] for every requested $s$; the semigroup property extends continuity from zero to other times. For smooth data, termwise derivatives converge and recover the original <Cauchy problem for a partial differential equation> uniquely.
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