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

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→+∞t→+∞. An estimate on the convergence rate is also given.