Understanding the Evolution of Tumours, a Phase-field Approach: Analytic Results and Optimal Control

Over the last decades, great strides have been made by the mathematical and medical communities towards the understanding of tumor growth. The recently achieved novelties arise from two leading factors: on the one hand, the flourishing of mathematical models for biological systems, and on the other hand, the more and more accurate computational methods and numerical solvers rose in the last decades. Despite the deep and challenging aim of understanding the hidden mechanisms behind the disease, the scientists' factual goal is to provide robust methods that may help the practitioners suiting the best therapy to every single patient. In this sense, the mathematical approach to tumour growth models might bring new lymph and hope to this arduous journey. This thesis aims at contributing to this common effort by providing some mathematical insights for two classes of tumor growth models capturing cell-to-cell adhesion effects of local and nonlocal nature, respectively. The common denominator of the two models under consideration is the assumption that the tumour cells are submerged in a nutrient-rich environment which is the primary source of the nourishment for the tumorous cells: this is a reasonable assumption at least for young tumours (avascular tumours). This paradigm leads us to analyse four-species PDE systems (tumour cells, healthy cells, nutrient-rich concentration, nutrient-poor concentration) which couples a Cahn-Hilliard type equation with source term for the tumour with a reaction-diffusion equation for the surrounding nutrient. For both models, we provide a rich spectrum of mathematical results. In the first part of the thesis, we discuss the weak well-posedness of the models under very general frameworks. Then, postulating further natural assumptions, we establish the strong well-posedness of the systems which lays the groundwork for possible further investigations. Lastly, we perform some singular limit analysis as some of the coefficients appearing in the systems approach zero. In the second part, based on the analytic results presented in the first one, we discuss some optimal control problems in which the governing equations are ruled by the previously analysed systems. In this direction, we provide the existence of minimisers and first-order necessary conditions for optimality. Via suitable asymptotic approaches, we then investigate the optimal control problems for the aforementioned systems as some parameters occurring in the systems go to zero.