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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 isHere $a=0.1\mathrm{c}{\mathrm{m}}^2$ and $t=10\mathrm{s}$, giving a width parameter of $0.5\mathrm{c}{\mathrm{m}}^2$ 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.

The uniform source adds a background without changing the spatial gradients relative to the unforced solution:At $10\mathrm{s}$ the offset is $1-e^{-1}\simeq 0.6321$. On the corrected grid the central peak is therefore about $0.8321$, and regions far from the hot spot are near $0.6321$. The hot spot's broadening is unchanged; the source is spatially uniform. The spatial mean of a periodic heat equation rises by the same amount, tending to a total increment of one temperature unit as $t\to \infty$.