  In this paper we consider an optimal control problem governed by a semilinear
heat equation with bilinear control-state terms and subject to control and
state constraints. The state constraints are of integral type, the integral
being with respect to the space variable. The control is multidimensional. The
cost functional is of a tracking type and contains a linear term in the control
variables. We derive second order necessary conditions relying on the concept
of alternative costates and quasi-radial critical directions. The appendix
provides an example illustrating the applicability of our results.
