Measure differential equation with a nonlinear growth/decay term

Abstract

In this paper we obtain an existence result for a measure differential equation with a nonlinear growth/decay term that may change the sign. This generalizes the model setting proposed by Piccoli and Rossi to applications in life sciences describing for example birth/death processes of individuals. The proof requires a modification of the approximating schemes introduced in the original paper of Piccoli and Rossi. The new scheme combines model discretization with an exponential solution of the nonlinear growth/decay term, and hence, preserves nonnegativity of the measure. Furthermore, we formulate a new analytic condition on the measure vector field, which substantially simplifies the previous proof of continuity of solutions with respect to initial data while simultaneously generalizing the former condition formulated by Piccoli and Rossi.