The existence and uniqueness of the solution of the nonlinear initial-boundary value problem
is established in for all and nonnegative initial data . This problem models the evolution of certain properties in populations of social organisms. Departing from well-known properties of -solutions of the corresponding linearized problem, the Riesz-Schauder fixed point theorem is applied to solve the above nonlinear problem for nonnegative initial data . Deriving some estimates depending only on the norm of and applying approximation by nonnegative initial data in as well as a compactness lemma, the existence result is extended to arbitrary nonnegative initial data.