Refined criteria toward boundedness in an attraction–repulsion chemotaxis system with nonlinear productions

Abstract

We study some zero-flux attraction-repulsion chemotaxis models, with nonlinear production rates for the chemorepellent and the chemoattractant, whose formulation can be schematized as (⋄) In this problem, Ω is a bounded and smooth domain of , for , , , reasonably regular functions generalizing, respectively, the prototypes and , for some and all . Moreover, and have specific expressions, and . Once for any sufficiently smooth , and the local well-posedness of problem () is ensured, and we establish for the classical solution defined in that the life span is indeed and u, v and w are uniformly bounded in in the following cases: For , , , and , provided (I.1) k<l; (I.2) ; (I.3) k = l and , or and . For , , , and , whenever (II.1) ; (II.2) and , or and ; (II.3) . For and and , under the assumptions k<l or (I.3)). In particular, in this paper we partially improve what derived in Viglialoro [Influence of nonlinear production on the global solvability of an attraction-repulsion chemotaxis system. Math Nachr. 2021;294(12):2441–2454] and solve an open question given in Liu and Li [Finite-time blowup in attraction-repulsion systems with nonlinear signal production. Nonlinear Anal Real World Appl. 2021;61:Paper No. 103305, 21]. Finally, the research is complemented with numerical simulations in bi-dimensional domains.