= Solution
Diffusion smooths and broadens the hot spot while conserving its integral. For an isolated two-dimensional initial profile $u_0=e^{-r^2/a}$, the <Gaussian hot spot under diffusion> is
$$
u(r,t)=\frac{a}{a+4\alpha t}\exp\left[-\frac{r^2}{a+4\alpha t}\right].
$$
Here $a=0.1\,\mathrm{cm^2}$ and $t=10\,\mathrm s$, giving \b[a width parameter of $0.5\,\mathrm{cm^2}$ and a peak of $0.20$] in the initial temperature units. The periodic-grid calculation agrees to displayed precision because the periodic copies are far away. Running the endpoint-inclusive listing as written instead gives a sampled initial peak near $0.9695$ and final peak near $0.1963$; the difference is consistent with the coordinate-grid issue, rather than a different diffusion law.
Increasing positive <thermal diffusivity> at fixed time produces a wider, lower peak. Each nonconstant <Fourier mode> decays faster, while the <spatial mean of a periodic heat equation> remains fixed. The profiles approach a uniform mean at long times under these <periodic boundary conditions>.
\Image[/past-exam-of-the-year-2-of-the-earth-sciences-geology-course-of-the-university-of-oxford/2025/paper-3-q7-diffusion-and-uniform-heating.png]
{title=Diffusion of a Gaussian hot spot, with and without a spatially uniform decaying heat source}
{description=Original calculations on an endpoint-excluded periodic grid. The cross-sections use a common temperature scale; the source contribution is a uniform offset. The dashed initial profile and the two final profiles are calculated rather than copied from the supplied examination figure.}
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