| Abstract: |
| We consider a non-local tumour growth model of phase-field type, describing the evolution of tumour cells through proliferation in the presence of a nutrient. The model consists of a coupled system, incorporating a non-local Cahn-Hilliard equation for the tumour phase variable and a reaction-diffusion equation for the nutrient. Non-local cell-to-cell adhesion effects are included through a convolution operator with appropriate spatial kernels. First, we establish novel regularity results for such a model, by applying maximal regularity theory in weighted $L^p$ spaces. Such a technique enables us to prove the local existence and uniqueness of a regular solution, including also chemotaxis effects. By leveraging time-regularisation properties and global boundedness estimates, we further extend the solution to a global one. These results provide the foundation for addressing an optimal control problem, aimed at identifying a suitable therapy, which can guide the tumour towards a predefined target. Specifically, we prove the existence of an optimal therapy and, by studying the Fr\`echet-differentiability of the control-to-state operator and introducing the adjoint system, we derive first-order necessary optimality conditions. |
|