Influence of nonlinear production on the global solvability of an attraction-repulsion chemotaxis system

This paper is dedicated to the attraction-repulsion chemotaxis-system (Formula presented.) (Formula presented.) defined in Ω, a smooth and bounded domain of (Formula presented.), with (Formula presented.). Moreover, (Formula presented.) and (Formula presented.) are suitably regular functions generalizing, for (Formula presented.) and α, (Formula presented.) the prototypes (Formula presented.), (Formula presented.), and (Formula presented.), (Formula presented.). We focus our analysis on the value (Formula presented.), establishing the temporal interval of existence of solutions (Formula presented.) to problem ((Formula presented.)). When zero-flux boundary conditions are fixed, we prove the following results, all excluding chemotactic collapse scenarios under certain correlations between the attraction and repulsive effects describing the model. To be precise, for every (Formula presented.), and (Formula presented.) (resp. (Formula presented.)), there exists (Formula presented.) (resp. (Formula presented.)) such that if (Formula presented.) (resp. (Formula presented.)), any sufficiently regular initial datum (Formula presented.) (resp. (Formula presented.) enjoying some smallness assumptions) produces a unique classical solution (Formula presented.) to problem ((Formula presented.)) which is global, i.e. (Formula presented.), and such that u, v and w are uniformly bounded. Conversely, the same conclusion holds true for every (Formula presented.), (Formula presented.), (Formula presented.) and any sufficiently regular (Formula presented.). Further, in a remark of the manuscript, we also address an open question posed in [21].