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

## TL;DR

This paper proves the existence and uniqueness of global classical solutions for a complex two-species cancer invasion model involving haptotaxis, cell proliferation, and extracellular matrix dynamics.

## Contribution

It establishes the first rigorous mathematical proof of global solutions for a multi-species cancer invasion model with epithelial-mesenchymal transition.

## Key findings

- Positivity of solutions is maintained.
- Global existence and uniqueness are proven under large initial data.
- The model captures key biological processes like cell migration and ECM remodeling.

## Abstract

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 positivity and conditional global existence and uniqueness of the classical solutions of the problem for large initial data.