Local asymptotics and optimal control for a viscous Cahn–Hilliard–Reaction–Diffusion model for tumor growth

¹ Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstrasse 8–10, 1040 Vienna, Austria
² Dipartimento di Matematica, Università di Pavia and IMATI – C.N.R., Via Ferrata 5, 27100 Pavia, Italy
³ Department of Mathematics, Politecnico di Milano, Via E. Bonardi 9, 20133 Milano, Italy
⁴ Department of Mathematics and Scientific Computing, University of Graz, Heinrichstraße 36, 8010 Graz, Austria

^* Corresponding author: lara.trussardi@uni-graz.at

Received: 6 December 2023
Accepted: 17 February 2025

Abstract

In this paper, we study nonlocal-to-local asymptotics for a tumor-growth model coupling a viscous Cahn–Hilliard equation describing the tumor proportion with a reaction–diffusion equation for the nutrient phase parameter. First, we prove that solutions to the nonlocal Cahn–Hilliard system converge, as the nonlocality parameter tends to zero, to solutions to its local counterpart. Second, we provide first-order optimality conditions for an optimal control problem on the local model, accounting also for chemotaxis, and both for regular or singular potentials, without any additional regularity assumptions on the solution operator. The proof is based on an approximation of the local control problem by means of suitable nonlocal ones, and on proving nonlocal-to-local convergence both for the corresponding dual systems and for the associated first-order optimality conditions.