ALEX 2018 Workshop: Abstracts

Recent advances in diffuse interface tumor growth analysis

Elisabetta Rocca

Mathematical Department, University of Pavia (Italy)

We consider the problem of long-time behavior of solutions and optimal control for a diffuse interface model of tumor growth. The state equations couples a Cahn-Hilliard equation and a reaction-diffusion equation, which models the growth of a tumor in the presence of a nutrient and surrounded by host tissue. The introduction of drugs into the system through the nutrient serves to eliminate the tumor cells, hence, in this setting the control variable will act on the nutrient equation. Furthermore, we allow the objective functional to depend on a free time variable, which represents the unknown treatment time to be optimized. As a result, we obtain first order necessary optimality conditions for both the drug concentration and the treatment time. One of the main aim of the control problem is to realize in the best possible way a desired final distribution of the tumor cells which is expressed by a target function that can be taken as a stable configuration of the system, so that the tumor does not grow again once the treatment is completed. In view of this fact we consider here also the problem of long-time behavior of solutions.

This is a joint project with C. Cavaterra (University of Milan), A. Miranville (University of Poitiers), G. Schimperna (University of Pavia), H. Wu (Fudan University, Shanghai).

Acknowledgments: This research has been performed in the framework of the project Fondazione Cariplo-Regione Lombardia MEGAsTAR “Matematica d’Eccellenza in biologia ed ingegneria come acceleratore di una nuova strateGia per l’ATtRattività dell’ateneo pavese” and by the Italian Ministry of Education, University and Research (MIUR): Dipartimenti di Eccellenza Program (2018–2022) - Dept. of Mathematics “F. Casorati”, University of Pavia.