Optimal purchase and advertising for a product with immediate sales start

We consider the problem of maximizing the discounted net profit of a firm which purchases a quantity of some product at a given time and afterwards advertises and sells the product progressively. We distinguish among the three possibilities of assuming the final time to be either fixed, or bounded, or free. In all cases, after stating the problem in the optimal control theory framework, we prove the existence of an optimal solution and characterize it using the Maximum Principle necessary conditions. Furthermore, we prove that the convexity of the purchase cost function is a sufficient condition for the uniqueness of the optimal solution.