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December  2018, 23(10): 4397-4431. doi: 10.3934/dcdsb.2018169

## Existence and uniqueness of global classical solutions to a two dimensional two species cancer invasion haptotaxis model

 1 Institute of Applied Analysis and Numerical Simulation, University of Stuttgart, 70569, Stuttgart, Germany 2 Institute of Mathematics, Johannes Gutenberg-University, 55128, Mainz, Germany 3 Institute of Applied Mathematics, Heidelberg University, 69120, Heidelberg, Germany

Received  April 2017 Revised  January 2018 Published  June 2018

Fund Project: J.G. was supported by the Baden-W¨urttemberg Stiftung via the project "Numerical Methods for Multiphase Flows with Strongly Varying Mach Numbers". N.K. was supported by the MaxPlanck Graduate School and the Impuls Fond "Single Cell" of the University of Mainz, M.L. was partially supported by the German Science Foundation (DFG) under the grant TRR 146 "Multiscale Simulation Methods for Soft Matter Systems". N.S was partially supported by the German Science Foundation (DFG) under the grant SFB 873 "Maintenance and Differentiation of Stem Cells in Development and Disease"

We consider a haptotaxis cancer invasion model that includes two families of cancer cells. Both families migrate on the extracellular matrix and proliferate. Moreover the model describes an epithelial-to-mesenchymal-like transition between the two families, as well as a degradation and a self-reconstruction process of the extracellular matrix.

We prove in two dimensional space positivity and conditional global existence and uniqueness of the classical solutions of the problem for large initial data.

Citation: Jan Giesselmann, Niklas Kolbe, Mária Lukáčová-Medvi${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over d} }}$ová, Nikolaos Sfakianakis. Existence and uniqueness of global classical solutions to a two dimensional two species cancer invasion haptotaxis model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4397-4431. doi: 10.3934/dcdsb.2018169
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##### References:
Graphical description on the model (1.1). The more aggressive CSCs escape the main body of the tumour and invade the ECM faster than the DCCs. At the same time, cancer secreted MMPs degrade the ECM
Simulation results of the Experiment 1. Showing here the spatial distribution of the DCC and CSC densities at several time instances. The CSCs invade the ECM by forming smooth "islands" that merge and smear-out further with time. On the other hand, the evolution of the DCCs is mostly diffusion dominated
Numerical simulation of the Experiment 2. Showing here the spatial distribution of the densities of the DCCs, CSCs components at several time instances. As opposed to Experiment 1, the particular choice of parameters leads to a highly dynamic invasion of the ECM by the CSCs. Thin waves interact and lead to complex invasion landscape
Butcher tableaux for the explicit (upper) and the implicit (lower) parts of the third order IMEX scheme (6.4), see also [21]
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