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 u, which migrates in response to a chemical signal v, within an impermeable habitat. Mathematically, this leads to the study of the following initial-boundary value problem (Formula presented) where (Formula presented), and (Formula presented) is a bounded and smooth domain of (Formula presented.) ((Formula presented.)) with boundary (Formula presented.), oriented by the outward unit normal vector (Formula presented.), and (Formula presented.). Moreover, for (Formula presented.), (Formula presented.), the function (Formula presented.) is assumed to take one of the following forms: (Formula presented.). Finally, (Formula presented.) 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 (Formula presented.), and both (Formula presented.) and (Formula presented.) remain uniformly bounded for all time, in the following situations: 1.For (Formula presented.), under the assumption (Formula presented.), whenever either (Formula presented.) and (Formula presented.), or (Formula presented.) and (Formula presented.); 2.For (Formula presented.), in these scenarios: (Formula presented.) and (Formula presented.); (Formula presented.), (Formula presented.) and (Formula presented.); and finally (Formula presented.), (Formula presented.) and (Formula presented.) for some (Formula presented.). This work builds upon previous studies that established global existence and boundedness results for variants of system ((Formula presented.)), in which the second equation takes the form (Formula presented.) (cf. [7] for the case (Formula presented.) and [11] for (Formula presented.)). In those settings, the source term (Formula presented.) typically has a more ad hoc structure, namely (Formula presented.). 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 ((Formula presented.)) and ((Formula presented.)). In the first case, the model structure ensures the control of the total mass (Formula presented.) over time, whereas in the second such control is no longer guaranteed, leading to substantially different qualitative dynamics.