Well-posedness and long-time behavior of a non-autonomous Cahn-Hilliard-Darcy system with mass source modeling tumor growth

TL;DR

This paper analyzes a non-autonomous Cahn-Hilliard-Darcy system with a mass source modeling tumor growth, proving existence, long-term behavior, and convergence to steady states in 2D and 3D.

Contribution

It establishes global existence of solutions, the existence of a minimal pullback attractor, and convergence results for the tumor growth model with non-autonomous sources.

Findings

Global weak and local strong solutions are proven to exist.

A minimal pullback attractor is established for the system.

Solutions converge to steady states when the source is asymptotically autonomous.

Abstract

In this paper, we study an initial boundary value problem of the Cahn-Hilliard-Darcy system with a non-autonomous mass source term $S$ that models tumor growth. We first prove the existence of global weak solutions as well as the existence of unique local strong solutions in both 2D and 3D. Then we investigate the qualitative behavior of solutions in details when the spatial dimension is two. More precisely, we prove that the strong solution exists globally and it defines a closed dynamical process. Then we establish the existence of a minimal pullback attractor for translated bounded mass source $S$. Finally, when $S$ is assumed to be asymptotically autonomous, we demonstrate that any global weak/strong solution converges to a single steady state as $t\to+\infty$. An estimate on the convergence rate is also given.