Solvability and Boundedness in Chemotaxis‐Consumption Models With Oppositely Acting Nonlocal Sources

In this paper, we investigate a chemotaxis system featuring a class of external source terms that incorporate both local and nonlocal growth as well as dampening effects. The model describes the evolution of a cell density , which migrates in response to a chemical signal , within an impermeable habitat. Mathematically, this leads to the study of the following initial‐boundary value problem where , and is a bounded and smooth domain of () with boundary , oriented by the outward unit normal vector , and . Moreover, for , , the function is assumed to take one of the following forms: Finally, indicates the maximal instant of time up to which the corresponding solutions exist. We provide conditions ensuring the absence of aggregation phenomena over time. Specifically, we show that the maximal existence time satisfies , and both and remain uniformly bounded for all time, in the following situations: For , under the assumption , whenever either and , or and ; For , in these scenarios: and ; , and ; and finally , and for some . This work builds upon previous studies that established global existence and boundedness results for variants of system (), in which the second equation takes the form (cf. [7] for the case and [11] for ). In those settings, the source term typically has a more ad hoc structure, namely . Based on this framework, the present study focuses on distinguishing the different dynamical behaviors exhibited by the systems in terms of the reaction described in () and (). In the first case, the model structure ensures the control of the total mass over time, whereas in the second such control is no longer guaranteed, leading to substantially different qualitative dynamics.