Mass concentration in a nonlocal model of clonal selection

Self-renewal is a constitutive property of stem cells. Testing the cancer stem cell hypothesis requires investigation of the impact of self-renewal on cancer expansion. To understand better this impact, we propose a mathematical model describing dynamics of a continuum of cell clones structured by the self-renewal potential. The model is an extension of the finite multi-compartment models of interactions between normal and cancer cells in acute leukemias. It takes a form of a system of integro-differential equations with a nonlinear and nonlocal coupling, which describes regulatory feedback loops in cell proliferation and differentiation process. We show that such coupling leads to mass concentration in points corresponding to maximum of the self-renewal potential and the model solutions tend asymptotically to a linear combination of Dirac measures. Furthermore, using a Lyapunov function constructed for a finite dimensional counterpart of the model, we prove that the total mass of the solution converges to a globally stable equilibrium. Additionally, we show stability of the model in space of positive Radon measures equipped with flat metric. The analytical results are illustrated by numerical simulations

An existence result for the Leray-Lions type operators with discontinuous coefficients

In this paper we prove an existence result for Leray-Lions quasilinear elliptic operator with discontinuous coefficients. The idea of the proof is based on compactness results for the sequences of solutions to regularized problems obtained via the Compensated Compactness, Young measures, and Set-Valued Analysis tools

On renormalized solutions to elliptic inclusions with nonstandard growth

We study the elliptic inclusion given in the following divergence form \begin{align*} & -\mathrm{div}\, A(x,\nabla u) \ni f\quad \mathrm{in}\quad \Omega, & u=0\quad \mathrm{on}\quad \partial \Omega. \end{align*} As we assume that $f\in L^1(\Omega)$, the solutions to the above problem are understood in the renormalized sense. We also assume nonstandard, possibly nonpolynomial, heterogeneous and anisotropic growth and coercivity conditions on the maximally monotone multifunction $A$ which necessitates the use of the nonseparable and nonreflexive Musielak--Orlicz spaces. We prove the existence and uniqueness of the renormalized solution as well as, under additional assumptions on the problem data, its relation to the weak solution. The key difficulty, the lack of a Carath\'{e}odory selection of the maximally monotone multifunction is overcome with the use of the Minty transform

A nonlinear structured population model: Global existence and structural stability of measure-valued solutions

This paper is devoted to the study of the global existence and structural stability of measure-valued solutions to a nonlinear structured population model given in the form of a nonlocal first-order hyperbolic problem on positive real numbers. In distinction to previous studies, where the L^1 norm was used, we apply the flat metric, similar to the Wasserstein W^1 distance. We argue that stability using this metric, in addition to mathematical advantages, is consistent with intuitive understanding of empirical data. Structural stability and the uniqueness of the weak solutions are shown under the assumption about the Lipschitz continuity of the kinetic functions. The stability result is based on the duality formula and the Gronwall-type argument. Using a framework of mutational equations, existence of solutions to the equations of the model is also shown under weaker assumptions, i.e., without assuming Lipschitz continuity of the kinetic functions

Asymptotic behaviour of a structured population model on a space of measures

In this paper we consider a physiologically structured population model with distributed states at birth, formulated on the space of non-negative Radon measures. Using a characterisation of the pre-dual space of bounded Lipschitz functions, we show how to apply the theory of strongly continuous positive semigroups to such a model. In particular, we establish the exponential convergence of solutions to a one-dimensional global attractor