On a fully parabolic chemotaxis system with nonlinear signal secretion

Abstract

This paper is devoted to the following nonlinear chemotaxis system ut=Δu−∇⋅(S(u)∇v)+f(u),x∈Ω,t>0,τvt=Δv−v+g(u),x∈Ω,t>0,under homogeneous Neumann boundary conditions. Here Ω⊂R3 is a bounded domain with smooth boundary, but not necessarily convex; S(u) satisfies |S(u)|≤|χ|uq with some q>0 and χ∈R; f(u) satisfies f(u)≤a−buα with some constants a≥0, b>0, α≥1; g(u) satisfies g(u)≤Kuγ with some positive constants K and γ. The corresponding parabolic–elliptic simplification of this system with τ=0 has been considered by Galakhov et al. (2016). This paper mainly deals with the fully parabolic system with τ=1. We mainly study the effects of chemotactic aggregation, the logistic damping, as well as the signal secretion on boundedness of the solutions, which, in particular, extend the recent results of Winkler (2010, Comm. Partial Differential Equations) and Lin and Mu (2016), as well as Xiang (2018, J. Math. Anal. Appl.), etc.