  Let $\mathfrak{a}$ be an ideal of a local ring $(R, \mathfrak{m})$ with $c =
\mathrm{cd}(\mathfrak{a},R)$ the cohomological dimension of $\mathfrak{a}$ in
$R$. In the case that $c=\dim R$, we first give a bound for
depth~$D(H^c_\mathfrak{a}(R))$, where $c>2$ and $(R,\mathfrak{m})$ is complete.
Later, $H^c_\mathfrak{a}(R) \otimes_R H^c_\mathfrak{a}(R)$,
$D(H^c_\mathfrak{a}(R)) \otimes_R D(H^c_\mathfrak{a}(R))$ and
$H^c_\mathfrak{a}(R) \otimes_R D(H^c_\mathfrak{a}(R))$ are examined. In the
case $c=\dim R-1$, the set Att$_R H^c_\mathfrak{a}(R)$ is considered.
