Temperature and damage in continuum models for tumor growth: analytic results and optimal control

Tumor growth is a complex phenomenon, driven by the interplay of genetic, biochemical, and mechanical factors, and its understanding remains a major challenge in modern medicine. Mathematical oncology offers a powerful quantitative framework to unravel the underlying mechanisms and to pave the way toward personalized treatments. This thesis falls within this context, introducing systems of partial differential equations (PDEs) intended to capture the fundamental biological and physical processes involved in cancer progression. Starting from well-established models that couple tumor dynamics with nutrient diffusion, we incorporate the evolution of additional relevant quantities, thereby enriching both the mathematical structure and the modeling viewpoint. Three mathematical models are introduced. The first is a nonisothermal phase field system of Caginalp type, designed to describe tumor progression under thermal therapies. Despite the biological relevance of temperature, this aspect has received limited attention in the literature; the approach proposed here provides a novel perspective and a solid foundation for future developments. Well-posedness for the related initial-boundary value problem and additional regularity are proven. In particular, the existence of a weak solution is demonstrated through a two-step approximation procedure, involving regularization of the potential and a Faedo-Galerkin discretization scheme. The second model builds on a Cahn–Hilliard-type system already incorporating mechanical effects—reflecting the viscoelastic properties of tissues—by introducing surgery-induced damage, a complete novelty in this field. In addition to the usual technical difficulties, such as the lack of mass conservation, this system presents further challenges due to the nonlinear coupling between the equations, in particular between the balance of forces and the differential inclusion governing the damage variable. The latter is by itself highly nonlinear, presenting both a p-Laplacian operator and the subdifferential of a nonsmooth convex potential. The existence of weak solutions is established through a carefully chosen time-discretisation scheme. However, due to the high nonlinearity, uniqueness remains an open problem. Nevertheless, if a suitable non-degenerative operator replaces the p-Laplacian, well-posedness can be obtained, giving new insights for further research. The third model is tailored to brain tumors and integrates lactate metabolism, viscoelasticity, and tissue damage, together with the action of cytotoxic and lactate-targeting drugs. Here, the evolution of the tumor is described through a Fisher–KPP-type equation. The existence of weak solutions is established using a Schauder fixed-point argument combined with a regularization of the potential. After proving additional regularity and uniqueness, an associated optimal control problem is considered, and the existence of optimal controls, together with their characterization via first-order necessary conditions, is demonstrated.