Cristian Morales-Rodrigo

ASYMPTOTIC BEHAVIOR OF GLOBAL SOLUTIONS TO A MODEL OF CELL INVASION

In this paper we analyze a mathematical model focusing on key events of the cells invasion process. Global well-possedness and asymptotic behaviour of nonnegative solutions to the corresponding coupled system of three nonlinear partial differential equations are studied.

Local existence and uniqueness of regular solutions in a model of tissue invasion by solid tumours

In this paper, we consider a nonlinear system of differential equations arising in tumour invasion which has been proposed in [M.A.J. Chaplain, A.R.A. Anderson, Mathematical modelling of tissue invasion, in: L. Preziosi (Ed.), Cancer Modelling and Simulation, Chapman & Hall/CRT, 2003, pp. 269-297]. The system consists of two PDEs describing the evolution of tumour cells and proteases, and an ODE which models the concentration of the extracellular matrix. We prove local existence and uniqueness of solutions in the class of Holder spaces. The proof of local existence is done by Schauders fixed point theorem, and for the uniqueness we use an idea from [H. Gajewski, K. Zacharias, Global behaviour of a reaction-diffusion system modelling chemotaxis, Math. Nachr. 195 (1998) 77-114].

GLOBAL EXISTENCE AND ASYMPTOTIC BEHAVIOR OF A TUMOR ANGIOGENESIS MODEL WITH CHEMOTAXIS AND HAPTOTAXIS

We consider a system of differential equations modeling tumor angiogenesis. The system consists of three equations: two parabolic equations with chemotactic terms to model endothelial cells and tumor angiogenesis factors coupled to an ordinary differential equation which describes the evolution of the fibronectin concentration. We study global existence of solutions and, under extra assumption on the initial data of the fibronectin concentration we obtain that the homogeneous steady state is asymptotically stable.

Long-time behavior of an angiogenesis model with flux at the tumor boundary

This paper deals with a nonlinear system of partial differential equations modeling a simplified tumor-induced angiogenesis taking into account only the interplay between tumor angiogenic factors and endothelial cells. Considered model assumes a nonlinear flux at the tumor boundary and a nonlinear chemotactic response. It is proved that the choice of some key parameters influences the long-time behavior of the system. More precisely, we show the convergence of solutions to different semi-trivial stationary states for different range of parameters.

A therapy inactivating the tumor angiogenic factors.

This paper is devoted to a nonlinear system of partial differential equations modeling the effect of an anti-angiogenic therapy based on an agent that binds to the tumor angiogenic factors. The main feature of the model under consideration is a nonlinear flux production of tumor angiogenic factors at the boundary of the tumor. It is proved the global existence for the nonlinear system and the effect in the large time behavior of the system for high doses of the therapeutic agent.