A nonlinear structured population model: Lipschitz continuity of measure-valued solutions with respect to model ingredients - Gwiazda, Piotr; Lorenz, Thomas; Marciniak-Czochra, Anna (Journal of differential equations, 2010)
This paper is devoted to the analysis of measure-valued solutions to a nonlinear structured population model given in the form of a nonlocal first-order hyperbolic problem on R+. We show global existence and Lipschitz continuity with respect to the model ingredients. In distinction to previous studies, where the L1 norm was used, we apply the flat metric, similar to the Wasserstein W1 distance. We argue that analysis using this metric, in addition to mathematical advantages, is consistent with intuitive understanding of empirical data. Lipschitz continuous dependence with respect to the model coefficients and initial data and the uniqueness of the weak solutions are shown under the assumption on the Lipschitz continuity of the kinetic functions. The proof of this result is based on the duality formula and the Gronwall-type argument.