Let
$$
\mathcal{L}\equiv\sum_{i,j}^{n}a_{i,j}\frac{\partial^{2}}{\partial x_{i}\partial x_{j}}+\sum_{i}^{n}b_{i}\frac{\partial}{\partial x_{i}}+c
$$
be uniformly elliptic on $\Omega\equiv\mathbb{R}^{n}\times\left(0,T\right)$.
Let
$$
\mathcal{M\equiv L}-\frac{\partial}{\partial t}
$$
Consider the parabolic equality
$$
\mathcal{M}u=f\text{ on }\Omega.
$$
Lastly, suppose $u\left(\mathbf{x},0\right)$ is monotone increasing
in $\mathbf{x}$ and that $f\left(\mathbf{x},t\right)$ is monotone
decreasing in $\mathbf{x}$ for all times $t$.
A function $g:\mathbb{R}^n \rightarrow \mathbb{R}$ is monotone increasing if for all $\mathbf{x},\mathbf{y}\in\mathbb{R}^n$, $g\left(\mathbf{x}\right)\leq g\left(\mathbf{y}\right)$ whenever $\mathbf{x}\leq \mathbf{y}$. Monotone decreasing is defined similarly.

First, consider the case of $a_{i,j}$, $b_{i}$, $c$ being functions of $t$ only. Define $v\left(\mathbf{x},t\right)=u\left(\mathbf{x}-\epsilon,t\right)$ (where $\epsilon \in \mathbb{R}^n$ with $\epsilon > 0$)
and note that
$$
\mathcal{M}v=f\left(\mathbf{x}-\epsilon,t\right)\text{ on }\Omega.
$$
Let $w=u-v$ so that 
$$
\mathcal{M}w=f\left(\mathbf{x},t\right)-f\left(\mathbf{x}-\epsilon,t\right)\leq0\text{ on }\Omega.
$$
Since $u$ is monotone at time zero, we have $$w\left(\mathbf{x},0\right)=u\left(\mathbf{x},0\right)-v\left(\mathbf{x},0\right)=u\left(\mathbf{x},0\right)-u\left(\mathbf{x}-\epsilon,0\right)\geq0 \text{ on } \mathbb{R}^n.$$
These last  two inequalities yield $w\geq0$ (and hence $u$ is monotone in $\mathbf{x}$) everywhere in $\Omega$.

However, this argument is rather restrictive, as it requires the assumption
that the coefficients $a_{i,j}$, $b_{i}$, and $c$ are constant in space (i.e. w.r.t. $\mathbf{x}$). Does anyone know of any relaxations to the above?