Optimal treatment for a phase field system of Cahn-Hilliard type modeling tumor growth by asymptotic scheme

Abstract

We consider a particular phase field system which physical context is that of tumor growth dynamics. The model we deal with consists of a Cahn-Hilliard type equation governing the evolution of the phase variable which takes into account the tumor cells proliferation in the tissue coupled with a reaction-diffusion equation for the nutrient. This model has already been investigated from the viewpoint of well-posedness, long time behavior, and asymptotic analyses as some parameters go to zero. Starting from these results, we aim to face a related optimal control problem by employing suitable asymptotic schemes. In this direction, further assumptions have to be required. Mainly, we ought to impose some quite general growth conditions for the involved potential and a smallness restriction for a parameter appearing in the system we are going to face. We provide existence of optimal controls and a necessary condition that an optimal control has to satisfy has been characterized as well.

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• Well-posedness and regularity for a fractional tumor growth model
• Strong well-posedness and inverse identification problem of a non-local phase field tumor model with degenerate mobilities
• Existence of weak solutions to multiphase Cahn-Hilliard-Darcy and Cahn-Hilliard-Brinkman models for stratified tumor growth with chemotaxis and general source terms
• Vanishing parameter for an optimal control problem modeling tumor growth
• Optimal control problems with sparsity for phase field tumor growth models involving variational inequalities