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.