On $\Bbb R^n$, the heat kernel and its derivatives vanish at infinity. Similarly, on compact manifolds without boundary we may say that the same thing happens (trivially, by saying that the heat kernel vanishes on the boundary). Do we understand why this happens? Can one say that this is true in general? In particular, understanding what happens on the following types of manifolds would help me gain a better intuition of the phenomenon:
- (compact) manifolds with non-empty boundary;
- bounded open subsets of $\Bbb R^n$ with smooth boundary;
- open subsets of $\Bbb R^n$ that are bounded in some directions and unbounded in others, such as $\{(x,y) \in \Bbb R^2 \mid x \ge 0 \}$ or $\{(x,y) \in \Bbb R^2 \mid x^2 \le y\}$ or $\{(r,\theta) \in \Bbb R^2 \mid r < 1 , \theta \in (\epsilon, 2\pi - \epsilon)\}$ with $\epsilon \ge 0$ small (a round pizza without a narrow slice (possibly a line segment), in polar coordinates).
Thank you.
