= Heat evolution of bounded data need not converge at large times
Bounded smooth initial data alone do not guarantee a pointwise large-time limit for the <heat equation> on Euclidean space. In four dimensions, $g(x)=(1+|x|^2)^i$ has modulus one. At the origin its <heat kernel> average is asymptotic to $4^i\Gamma(2+i)t^i$, which oscillates indefinitely. Initial data vanishing at infinity do have heat averages tending to zero. This distinction matters when using <Duhamel principle> to construct a stationary solution for a time-independent source.
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