Mathematical models for tumour growth are becoming increasingly common in the recent scientific literature, as their main aim is the development of patient-specific models that can help clinicians' decisions through personalised tumour forecasts. Here, we consider models of phase-field type, mainly concerning young avascular tumours. Thus, we describe a tumour through a phase variable, representing the difference in volume fractions between cancerous cells and healthy cells in a given tissue. More specifically, our models are systems of PDEs of Cahn-Hilliard type, coupled to additional reaction-diffusion equations for other key quantities, such as nutrients used by the tumour cells to proliferate. During the talk, we will present some recent results and challenges in the mathematical analysis of such models. In particular, we will focus on the well-posedness of the PDE systems and the regularity of their solutions, as well as problems related to optimal control of therapies and inverse reconstruction of earlier states.
