Let $M:=\mathbb{R}^n\setminus \mathbb{D}^n$, where $\mathbb{D}^n$ is the open unit ball in $\mathbb{R}^n$ and $u\colon M\times [0,\infty)\rightarrow \mathbb{R}$ is solution of the following PDE \begin{equation}\label{heat 1} \begin{split} \frac{\partial u}{\partial t}&=\Delta u\\ u(x,0)&=u_0(x). \end{split} \end{equation}
Question 1: Is there any parabolic maximum principle:$$ \sup_{M\times[0,\infty)} u(x,t)= \sup_{M\times\{0\} \cup \partial M\times[0,\infty)} u(x,t),$$ if $$\sup_{M\times\{0\} \cup \partial M\times[0,\infty)} u(x,t)<\infty\text{?} $$
Question 2: Is there a similar kind of parabolic maximum principle for non-compact manifold with boundary?
