Boundedness in a fully parabolic chemotaxis system with logistic-type source and nonlinear production. (June 2019) - Zhuang, Mengdi; Wang, Wei; Zheng, Sining
Abstract: In this paper we study the fully parabolic chemotaxis system with logistic-type source and nonlinear production: u t = Δ u − χ ∇ ⋅ ( u ∇ v ) + f ( u ), v t = Δ v − v + g ( u ), subject to the non-flux boundary conditions with a smooth and bounded domain Ω ⊂ R n ( n ≥ 1 ), and the nonnegative initial data u 0 ∈ C 0 ( Ω ̄ ) and v 0 ∈ W 1, ∞ ( Ω ), where the sensitivity χ > 0, the logistic-type source f ( s ) ≤ s − μ s α for s ≥ 0 with α > 1, μ > 0, and the nonlinear production g ( s ) ≤ s ( s + 1 ) β − 1 for s ≥ 0 with β > 0 . It is obtained that the problem possesses a globally bounded classical solution if the self-restriction mechanism of the logistic-type source dominates the nonlinear production that β < α − 1, or β = α − 1 with μ > 0 sufficiently large. This extends the current global existence result by Nakaguchi and Osaki (2018). The situation with the nonlinear production controlled by the linear diffusion that β ∈ ( 0, 2 n ) is considered as well, which is parallel to the model with nonlinear diffusion and subcritical chemotactic sensitivity without logistic source studied by Tao and Winkler (2012).