We study a Cahn–Hilliard-type model for tumor growth that is coupled to an additional evolution equation describing the concentration of a chemical species influencing the phase-separation process. The dynamics of this concentration variable are governed by a nonlinear parabolic equation containing a cross-diffusion term reminiscent of those appearing in Keller–Segel-type chemotaxis systems. In contrast to the classical Keller–Segel model, we establish the existence of global weak solutions for the coupled system.
Beyond the standard weak formulation, we introduce an extended solution concept that incorporates a logarithmic inequality analogous to an entropy inequality. We discuss the respective roles of these ingredients in proving weak–strong uniqueness and in establishing the sequential closedness of the solution set.