Regularity of multipliers and second-order optimality conditions for semilinear parabolic optimal control problems with mixed pointwise constraints

A class of optimal control problems governed by semilinear parabolic equations with mixed pointwise constraints is considered. We give some criteria under which the first and second-order optimality conditions are of KKT-type. We then prove that the Lagrange multipliers belong to $L^p$-spaces. Moreover, we show that if the initial value is good enough and boundary $\mathrm{\partial }\mathrm{\Omega }$ has a property of positive geometric density, then multipliers and optimal solutions are Hölder continuous.