Boundedness in a chemotaxis system with indirect signal production and generalized logistic source

Abstract

This work considers the chemotaxis-growth system u t = Δ u − ∇ ⋅ ( u ∇ v ) + u − u α , v t = Δ v − v + w , w t = u − δ w , in a smoothly bounded domain Ω ⊂ R n , n ≥ 2 , with zero-flux boundary conditions, where δ > 0 and α > 1 are given positive parameters. In the case when n = 3 and α = 2 , the global existence and boundedness of smooth solutions to this system was previously asserted in Hu and Tao (2016). Inspired by an approach newly developed in Tao and Winkler (2015), the present work improves the aforementioned result to a general case when n ≥ 2 and α > n 2 .