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

TL;DR

This paper analyzes the transition from nonlocal to local models in a viscous Cahn-Hilliard-Reaction-Diffusion system for tumor growth, establishing convergence results and deriving optimal control conditions.

Contribution

It proves nonlocal-to-local convergence of solutions and dual systems, and derives first-order optimality conditions for the local control problem without extra regularity assumptions.

Findings

Solutions to the nonlocal system converge to the local system as nonlocality vanishes.

First-order optimality conditions are established for the local control problem.

The approach handles both regular and singular potentials without additional regularity.

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.