 Bounded parabolic differentiability class

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.