On a Cahn–Hilliard–Keller–Segel model with generalized logistic source describing tumor growth

We propose a new type of diffuse interface model describing the evolution tumor mass under effects chemical substance (e.g., nutrient or drug). The process is described by utilizing variables ?, an order parameter representing local proportion cells, and ?, concentration chemical. ? assumed to satisfy suitable form Cahn–Hilliard equation with source logarithmic potential Flory–Huggins (or generalizations it). ? satisfies reaction-diffusion where cross-diffusion term has same expression as in celebrated Keller–Segel model. In this respect, we represents coupling between subsystem believe that, compared other models, choice more effective capturing chemotactic that may occur growth dynamics (chemically induced consumption nutrient/drug cells). Note prevent finite time blowup assume logistic type. Our main mathematical result devoted proving existence weak solutions rather general setting covers both two- three- dimensional cases. Under restrictive assumptions on coefficient data, some cases spatial dimension, prove various regularity results. Finally, proper class smooth show uniqueness continuous dependence initial data number significant