On a nonlinear diffussive model for the evolution of cells within a moving domain

We investigate the dynamics of a nonlinear model describing the motion of cells under the effect of porous-medium diffusion and transport and in the presence of nutrient and drug application. The momentum equation for the evolution of the velocity field is governed by Darcy’s law, while the evolution of the chemical attractant (nutrient or drug) is governed by a diffusion equation. The system evolves within a moving domain in R 3 R^3 accounting for the expansion or shrinkage of the tumor. The global existence of weak solutions is established with the aid of a regularized approximating scheme and an Arbitrary Lagrangian-Eulerian (ALE) mapping for the motion of the tumor. This work provides a variational framework suitable for both analysis and simulations.