  A class of optimal control problems governed by semilinear parabolic
equations with mixed constraints and a box constraint for control variable is
considered. We show that if the separation condition is satisfied, then both
optimality conditions of KKT-type and regularity of multipliers are fulfilled.
Moreover, we show that if the initial value is good enough and boundary
$\partial\Omega$ has a property of positive geometric density, then multipliers
and optimal solutions are H\"{o}lder continuous.
