Nagumo-type viability theorem for nonlocal balance equation

The main object of the paper is a nonlocal balance equation that describes an evolution of a system of infinitely many identical particles those move according to a vector field and can also disappear or give a spring. For such system we examine the viability property that means that the systems starting inside a given set of measures does not leave this set. We prove an analog of the Nagumo-type viability theorem that gives the equivalent form of the viability property in the terms of the tangent cone.