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 Hölder spaces. The proof of local existence is done by Schauder’s 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].