Some remarks on the continuity equation

We describe some relations between the properties of the Cauchy problem for an ODE and the properties of the Cauchy problem for the associated continuity equation in the class of measures

Conditions for the solvability of the Cauchy problem for linear first-order functional differential equations

Conditions for the unique solvability of the Cauchy problem for a family of scalar functional differential equations are obtained. These conditions are sufficient for the solvability of the Cauchy problem for every equation from the family and are necessary for the solvability of the Cauchy problem for all equations from the family. In contrast to many known articles, we consider equations with functional operators acting into the space of essentially bounded functions