Global existence and lower bounds in a class of tumor-immune cell interactions chemotaxis systems

This paper investigates the properties of classical solutions to a class of chemotaxis systems that model interactions between tumor and immune cells. Our focus is on examining the global existence and explosion of such solutions in bounded domains of $ \mathbb{R}^n $, $ n\geq3 $, under Neumann boundary conditions. We distinguish between two scenarios: one where all equations are parabolic and another where only one equation is parabolic while the rest are elliptic. Boundedness is demonstrated under smallness assumptions on the initial data in the former scenario, while no such constraints are necessary in the latter. Additionally, we provide estimates for the blow-up time of unbounded solutions in three dimensions, supported by numerical simulations.