Measure Differential Equation with a Nonlinear Growth/Decay Term

TL;DR

This paper establishes an existence result for a measure differential equation incorporating a nonlinear growth/decay term, introducing a novel approximation scheme that preserves nonnegativity and simplifies the analysis of solution continuity.

Contribution

It introduces a modified approximation scheme combining discretisation and exponential solutions to handle nonlinear growth/decay in measure differential equations.

Findings

Proves existence of solutions under new conditions.

Develops a scheme that preserves nonnegativity.

Simplifies the proof of solution continuity.

Abstract

We obtain an existence result for a Measure Differential Equation with an additional nonlinear growth/decay term that may change the sign. The proof requires a modification of approximating schemes proposed by Piccoli and Rossi. The new scheme combines model discretisation with the exponential solution of the nonlinear growth/decay and hence preserves nonnegativity of the measure. Furthermore, we formulate a simpler analytic condition on the Measure Vector Field which substantially simplifies the previous proof of continuity of solutions with respect to initial data.