= Solution
Differentiate the spatial <Fourier transform> under the integral, and integrate the second spatial derivative by parts twice. For sufficiently decaying temperature and first derivative, the boundary terms vanish:
$$
\partial_t\widetilde\theta(k,t)=\int_{\mathbb R}e^{-ikx}\partial_x^2\theta\,dx=(ik)^2\widetilde\theta(k,t).
$$
Hence the transformed <heat equation> is the <ordinary differential equation>
$$
\boxed{\partial_t\widetilde\theta=-k^2\widetilde\theta.}
$$
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