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doi: 10.3934/mcrf.2019040

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

 Dipartimento di Matematica e Applicazioni, Università di Milano–Bicocca, via Cozzi 55, 20125 Milano, Italy

Received  February 2019 Revised  May 2019 Published  August 2019

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 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, we assume 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 the existence of optimal controls and a necessary condition for optimality is addressed.

Citation: Andrea Signori. Optimal treatment for a phase field system of Cahn-Hilliard type modeling tumor growth by asymptotic scheme. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2019040
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