- We study the existence of weak solutions for a model of cell invasion into the extracellular matrix (ECM), consisting of a non-linear partial differential equation (PDE) for cell density coupled with an ordinary differential equation (ODE) for ECM density. The model includes cross-species density-dependent diffusion and proliferation terms, capturing the role of the ECM in supporting cells during invasion and preventing growth via volume-filling effects. The occurrence of cross-diffusion terms is a common theme in the system of interacting species with excluded-volume interactions. Additionally, ECM degradation by cells is included. We present an existence result for weak solutions, exploiting the partial gradient flow structure to overcome the non-regularising nature of the ODE. Furthermore, we present simulations that illustrate travelling wave solutions and investigate asymptotic behaviour as the ECM degradation rate tends to infinity.
