Analysis of a multi-species Cahn–Hilliard–Keller–Segel tumor growth model with chemotaxis and angiogenesis

We introduce a multi-species diffuse interface model for tumor growth, characterized by its incorporation of essential features related to chemotaxis, angiogenesis and proliferation mechanisms. We establish the weak well-posedness of the system within an appropriate variational framework, accommodating various choices for the nonlinear potentials. One of the primary novelties of the work lies in the rigorous establishment of the existence of a weak solution through the introduction of delicate approximation schemes. To our knowledge, this represents a novel advancement for both the intricate Cahn–Hilliard–Keller–Segel system and the Keller–Segel subsystem with source terms. Moreover, when specific conditions are met, such as having more regular initial data, a smallness condition on the chemotactic constant with respect to the magnitude of initial conditions and potentially focusing solely on the two-dimensional case, we provide regularity results for the weak solutions. Finally, we derive a continuous dependence estimate, which, in turn, leads to the uniqueness of the smoothed solution as a natural consequence.