Consider the 2nd-order uniformly elliptic linear parabolic PDE
$$\partial_tu = \Delta u+b\cdot \nabla u, \\ u(0,\cdot) = h$$
on $(0,T)\times \mathbb{R}^d$, with $b: (0,T)\times \mathbb{R}^d \to \mathbb{R}^d$ smooth with support contained in some $[s_1,s_2]\times K$ with $K \subseteq \mathbb{R}^d$ compact and $[s_1,s_2] \subseteq (0,T)$. Suppose also that $h \in C_c^{\infty}(\mathbb{R}^d)$ - i.e. in terms of the given data, the situation is as good as it gets.
Does there exist a (classical) solution $u$ to this equation, with $||u||_{\infty}, ||\partial_tu||_{\infty}, ||\nabla u||_{\infty}, ||D^2u||_{\infty}$ finite? Considering the convolution of the (existing) unique fundamental solution $Z$ with $h$, we obtain a bounded, classical solution with bounded spatial derivatives as desired. However, I am uncertain about $\partial_tu$: Calculating similarly as for the spatial derivatives (i.e. estimating the derivative of the fundamental solution by the corresponding derivative of the fundamental solution to the classical heat equation and using the boundedness of the initial data $h$) does not suffice and suggests a blowup for $t \to 0$ instead. However, one can clearly bound $\partial_tu$ on each $[\epsilon,T]\times \mathbb{R}^d$. Now I read a survey type article, which stated that under suitable conditions on $h$ (for which $h \in C_c^{\infty}$ suffices), one can show (among other properties) boundedness of $\partial_t u$ on any $[0,T] \times B_R(0)$. Combining these to aspects suggests that non-boundedness of $\partial_t u$ may only happen in some $[0,\epsilon] \times B_R(0)^c$. But intuitively, for $t$ close enough to $0$, $\partial_tu(tax)$ should decrease for increasing $|x|$. Hence, choosing a large enough ball $B_R(0)$ (which particularly contains the support of $h$) and $\epsilon$ small enough, I'd guess that one can bound $||\partial_tu||_{\infty}$on $[0,\epsilon]\times B_R(0)^c$ by its maximum on $[0,\epsilon]\times B_R(0)$, which is finite as mentioned above.
Is my reasoning correct and/or is it true that one can obtain boundedness of $||\partial_t u||_\infty$? I'm not experienced with things like Schauder estimates for parabolic equations, so I apologize, if this question is stupid or easy to answer in either direction.