Consider the heat equation $$ (\partial_t + \Delta + V)u = 0$$ on a complete (open) Riemannian manifold with bounded geometry, where $V$ is a smooth and bounded potential.

Consider the semigroup generated by $A:=\Delta + V$ on the spaces $X = C_0(M)$ or $X = L^p(M)$ for $1\leq p < \infty$. The domain of $A$ is in each case $$ \mathrm{dom}(A) =\{ u \in X \mid Au \in X \},$$ where $Au$ denotes the distributional derivative.

**My question is the following:** It is clear by parabolic regularity that $e^{-tA}u$ is smooth for any $u \in X$. But is it also true that solutions $e^{-tA}u$ satisfy
$$ |\nabla^ku| \in X ~~~~~~~\forall ~k$$
for $u \in \mathrm{dom}(A)$?
The local statement is clear (i.e. solutions are in $W^{k, p}_{\mathrm{loc}}(M)$ and in $C^k(M)$ for every $k$), but is it also true that we have this decay at infinity?

What about the special case of vanishing $V$?