= Solution
Apply the <Fourier transform of a derivative> twice in each spatial coordinate, and commute the time derivative with the spatial <Fourier transform>. For the <Laplacian>,
$$
\mathcal F[\Delta u]=-(k_x^2+k_y^2)\widehat u.
$$
Thus
$$
\boxed{\partial_t\widehat u=-\alpha(k_x^2+k_y^2)\widehat u.}
$$
The minus sign is essential: positive <thermal diffusivity> damps short spatial scales rather than amplifying them. The <zero Fourier mode> is unchanged in the absence of a source.
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