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A class of optimal control problems governed by semilinear parabolicequations with mixed constraints and a box constraint for control variable isconsidered. We show that if the separation condition is satisfied, then bothoptimality 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 multipliersand optimal solutions are H\"{o}lder continuous.

Regularity of multipliers and second-order optimality conditions of KKT-type for semilinear parabolic control problems