This work considers the chemotaxis-growth system [Formula: see text] in a smoothly bounded domain [Formula: see text], with zero-flux boundary conditions, where [Formula: see text] and [Formula: see text] are given positive parameters. In striking contrast to the corresponding three-dimensional two-component chemo-taxis-growth system to which the global existence or blow-up of classical solutions largely remains open when [Formula: see text] is small, it is shown that whenever [Formula: see text] [Formula: see text] and [Formula: see text], for any given non-negative and suitably smooth initial data [Formula: see text] satisfying [Formula: see text], the system (⋆) admits a unique global classical solution that is uniformly-in-time bounded, which rules out the possibility of blow-up of solutions in finite time or in infinite time. Moreover, under the fully explicit condition [Formula: see text] the solution [Formula: see text] exponentially converges to the constant stationary solution [Formula: see text] in the norm of [Formula: see text] as [Formula: see text].
Blow-up phenomena and global existence for the periodic two-component Dullin-Gottwald-Holm system