I am not too familiar with the theory of regularity of parabolic PDEs, but I am wondering whether the following is true:
Let $b:[0,T] \times \mathbb{R}^d \to \mathbb{R}^d$ be in $C([0,T], C^1_b)$. Then can we find some function $g: \mathbb{R} \to \mathbb{R}$ such that there exists a solution $U: [0,T] \times \mathbb{R}^d \to \mathbb{R}$ to the PDE $$ \frac{\partial U}{\partial t} + \frac{1}{2} \Delta U + b \cdot \nabla U = g \circ U,$$ where $\frac{\partial U}{\partial t}$ is continuous, $g \circ U$ is Lipschitz and the second order derivatives in space $ \frac{\partial^2 U}{\partial x_i \partial x_j}(t,\mathbf{x}) >0$, for $i,j \in \{1, \ldots, d\}$?
If the answer is positive, can we further replace the condition that $ \frac{\partial^2 U}{\partial x_i \partial x_j}(t,\mathbf{x}) >0$ by $ \Big| \frac{\partial^2 U}{\partial x_i \partial x_j}(t,\mathbf{x}) \Big| >M$, for $i,j \in \{1, \ldots, d\}$, for some $M>0$?
The main problem is that most of the regularity results in the literature only focus on results regarding boundedness or integrability of the derivatives, which is not what I want. Any ideas?