Boundedness properties of very weak solutions to a fully parabolic chemotaxis-system with logistic source

In this paper we study the chemotaxis-system {ut=Δu−χ∇⋅(u∇v)+g(u)x∈Ω,t>0,vt=Δv−v+ux∈Ω,t>0, defined in a convex smooth and bounded domain Ω of R3, with χ>0 and endowed with homogeneous Neumann boundary conditions. The source g behaves similarly to the logistic function and verifies g(s)≤a−bsα, for s≥0, with a≥0, b>0 and α>1. In line with Viglialoro (2016), where for α∈(53,2) the global existence of very weak solutions (u,v) to the system is shown for any nonnegative initial data (u0,v0)∈C0(Ω̄)×C2(Ω̄) and under zero-flux boundary condition on v0, we prove that no chemotactic collapse for these solutions may present over time. More precisely, we establish that if the ratio ab does not exceed a certain value and for 95<p<α<2 the initial data are such that ‖u0‖Lp(Ω) and ‖∇v0‖L4(Ω) are small enough, then (u,v) is uniformly-in-time bounded.