I am considering the asymptotic behaviour of the integral below over the unit cube
$$\int\limits_{[0,1]^k} |A(t) p| \operatorname{d}r^1\dots\operatorname{d}r^k $$
Where $p \in \mathbb{R}^n,\ A(t)\in \mathbb{R}^{n\times n}$ and $|\cdot|$ is some norm and $||\cdot||$ is the matrix norm induced by the vector norm $|\cdot|$. I know that $||A(t)||\leq \operatorname{e}^{-at} ||A(0)||$. Can I conclude that the integral above is strictly decreasing and that converges to $0$ as $t\to+\infty$? It seems extra trivial, but for some reason in the paper I am reading they only claim that the integral is decreasing with time.
What if instead I consider the integral below with $p(r)$ being a function?
$$\int\limits_{[0,1]^k} |A(t) p(r)| \operatorname{d}r^1\dots\operatorname{d}r^k $$
I would say that I need to include some extra conditions on the function $p(r)$. Would the continuity of $p(r)$ on $[0,1]^k$ be sufficient to conclude the same thing?