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We study the effects on $D$ of assuming that the power series ring $D[[X]]$ is a $v$-domain or a PVMD. We show that a PVMD $D$ is completely integrally closed if and only if $\bigcap_{n=1}^{\infty }(I^{n})_{v}=(0)$ for every proper $t$-invertible $t$-ideal $I$ of $D$. Using this, we show that if $D$ is an AGCD domain, then $D[[X]]$ is integrally closed if and only if $D$ is a completely integrally closed PVMD with torsion $t$-class group. We also determine several classes of PVMDs for which being Archimedean is equivalent to being completely integrally closed and give some new characterizations of integral domains related to Krull domains.