  This paper proposes an optimal control problem for a parabolic equation with
a nonlocal nonlinearity. The system is described by a parabolic equation
involving a nonlinear term that depends on the solution and its integral over
the domain. We prove the existence and uniqueness of the solution to the system
and the boundedness of the solution. Regularity results for the
control-to-state operator, the cost functional and the adjoint state are also
proved. We show the existence of optimal solutions and derive first-order
necessary optimality conditions. In addition, second-order necessary and
sufficient conditions for optimality are established.
