  In this paper, we shall introduce $h$-expansiveness and asymptotical
$h$-expansiveness for actions of sofic groups. By the definitions, each
$h$-expansive action of sofic groups is asymptotically $h$-expansive. We show
that each expansive action of sofic groups is $h$-expansive, and, for any given
asymptotically $h$-expansive action of sofic groups, the entropy function (with
respect to measures) is upper semi-continuous and hence the system admits a
measure with maximal entropy.
  Observe that asymptotically $h$-expansive property was firstly introduced and
studied by Misiurewicz for $\mathbb{Z}$-actions using the language of
topological conditional entropy. And thus in the remaining part of the paper,
we shall compare our definitions of weak expansiveness for actions of sofic
groups with the definitions given in the same spirit of Misiurewicz's ideas
when the group is amenable. It turns out that these two definitions are
equivalent in this setting.
