Does the size of the gradient decrease in this heat-type equation?

Question

Let $\Omega\subset\mathbb{R}^n$ and $T>0$. Suppose all functions are smooth and $u$ solves $$\frac{\partial u}{\partial t}(x,t)-\Delta u(x,t)+\underbrace{\sum_{i=1}^nf_i(x)\frac{\partial u}{\partial x_i}(x,t)+g(x)}_{\text{lower order terms}}=0,\quad (x,t)\in\Omega\times(0,T).$$ Do we always have that $$\frac{\partial}{\partial t}\lvert\nabla u(x,t)\rvert^2\leq0$$ for all $(x,t)\in\Omega\times(0,T)$ or are there counterexamples?