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

Signori Andrea

doi:10.3934/mcrf.2019040

Article Contents

2020, Volume 10, Issue 2: 305-331. Doi: 10.3934/mcrf.2019040

This issue Previous Article On the exact controllability and the stabilization for the Benney-Luke equation Next Article Finite element error estimates for one-dimensional elliptic optimal control by BV-functions

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

Signori Andrea

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

Received: February 2019

Revised: May 2019

Early access: August 2019

Published: April 2020

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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 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.

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Mathematics Subject Classification: Primary: 35K61, 35Q92, 49J20, 49K20; Secondary: 35K86, 92C50.

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