  We propose a novel approach to modeling advertising dynamics for a firm
operating over distributed market domain based on controlled partial
differential equations of diffusion type. Using our model, we consider a
general type of finite-horizon profit maximization problem in a monopoly
setting. By reformulating this profit maximization problem as an optimal
control problem in infinite dimensions, we derive sufficient conditions for the
existence of its optimal solutions under general profit functions, as well as
state and control constraints, and provide general characterization of the
optimal solutions. Sharper, feedback-form, characterizations of the optimal
solutions are obtained for two variants of the general problem.
