= Bounded parabolic differentiability class
{title2=$C_b^{1,2}$}
The notation $C_b^{1,2}([0,\infty)\times\mathbb R)$ ordinarily denotes functions with continuous time derivative and two continuous space derivatives, with the function and indicated derivatives bounded globally. A different condition, often intended in finite-horizon <partial differential equations>, is membership in $C_b^{1,2}([0,T]\times\mathbb R)$ for each finite $T$, allowing bounds to grow with $T$. These conventions must be distinguished in a long-time growth problem: the second permits divergence with time; the first does not. There is no universal notation for the finite-horizon condition, so it should be stated explicitly.
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