Let $\Omega\subset\mathbb{R}^3$ be a bounded smooth domain and consider the following initial boundary value problem \begin{align} \partial_tu-\Delta u+b\cdot\nabla u&=0\\ u_{|\partial\Omega}&=0\\ u(t=0)&=u_0\ge 0 \end{align} where $b$ is a given, time-independent, smooth, bounded vector field on $\Omega$.
Heuristically, transport by a smooth, bounded vector field preserves the $L^\infty$ norm of the solution whereas diffusion, due to the Laplacian, leads to decay. Then, can we infer that for the unique solution to the above problem, which is a combination of transport and diffusion, we necessarily have $u\to 0$ as $t\to\infty$ where the first limit is taken in any/some appropriate sense (uniform, $L^p$, whatever)? I'm inclined to believe that this is the case since we have the consequence of the parabolic strong maximum principle which implies that $M(t):=\max_x u(t,x)$ is strictly decreasing. But it doesn't follow immediately that $M(t)\to 0$, so wondering if there are some counterexamples...
Edit: if $b$ were divergence-free we could use a simple energy method argument. But here I'm not assuming that.