= Solution
The uniform source adds a background without changing the spatial gradients relative to the unforced solution:
$$
\boxed{u_{\rm forced}(x,y,t)=u_{\rm unforced}(x,y,t)+1-e^{-t/10}.}
$$
At $10\,\mathrm s$ the offset is \b[$1-e^{-1}\simeq0.6321$]. On the corrected grid the central peak is therefore \b[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 $t\to\infty$.
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