Influence of nonlinear production on the global solvability of an attraction‐repulsion chemotaxis system
Giuseppe Viglialoro
Abstract
Abstract This paper is dedicated to the attraction‐repulsion chemotaxis‐system defined in Ω, a smooth and bounded domain of , with . Moreover, and are suitably regular functions generalizing, for and α, the prototypes , , and , . We focus our analysis on the value , establishing the temporal interval of existence of solutions to problem (). 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 , and (resp. ), there exists (resp. ) such that if (resp. ), any sufficiently regular initial datum (resp. enjoying some smallness assumptions) produces a unique classical solution to problem () which is global, i.e. , and such that u , v and w are uniformly bounded. Conversely, the same conclusion holds true for every , , and any sufficiently regular . Further, in a remark of the manuscript, we also address an open question posed in [21].