Boundedness properties of very weak solutions to a fully parabolic chemotaxis-system with logistic source. (April 2017) - Viglialoro, G.
Abstract: In this paper we study the chemotaxis-system u t = Δ u − χ ∇ ⋅ ( u ∇ v ) + g ( u ) x ∈ Ω, t > 0, v t = Δ v − v + u x ∈ Ω, t > 0, defined in a convex smooth and bounded domain Ω of R 3, with χ > 0 and endowed with homogeneous Neumann boundary conditions. The source g behaves similarly to the logistic function and verifies g ( s ) ≤ a − b s α, for s ≥ 0, with a ≥ 0, b > 0 and α > 1 . In line with Viglialoro (2016), where for α ∈ ( 5 3, 2 ) the global existence of very weak solutions ( u, v ) to the system is shown for any nonnegative initial data ( u 0, v 0 ) ∈ C 0 ( Ω ̄ ) × C 2 ( Ω ̄ ) and under zero-flux boundary condition on v 0, we prove that no chemotactic collapse for these solutions may present over time. More precisely, we establish that if the ratio a b does not exceed a certain value and for 9 5 < p < α < 2 the initial data are such that ‖ u 0 ‖ L p ( Ω ) and ‖ ∇ v 0 ‖ L 4 ( Ω ) are small enough, then ( u, v ) is uniformly-in-time bounded.,