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 $t$.

First, consider the case of $a_{i,j}$, $b_{i}$, $c$ constant on $\Omega$. Define $v\left(\mathbf{x},t\right)=u\left(\mathbf{x}-\epsilon,t\right)$ 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. Does anyone know of any relaxations to the above?