On a fully parabolic chemotaxis system with nonlinear signal secretion

This paper is devoted to the following nonlinear chemotaxis system {u(t) = Delta u - del . (S(u)del v) + f(u), x is an element of Omega, t > 0, tau v(t) = Delta v - v + g(u), x is an element of Omega, t > 0, under homogeneous Neumann boundary conditions. Here Omega subset of R-3 is a bounded domain with smooth boundary, but not necessarily convex; S(u) satisfies vertical bar S(u)vertical bar <= vertical bar chi vertical bar u(q) with some q > 0 and chi is an element of R; f(u) satisfies f(u) <= a-bu(alpha) with some constants a >= 0, b > 0, alpha >= 1; g(u) satisfies g(u) < Ku(gamma) with some positive constants K and gamma. The corresponding parabolic elliptic simplification of this system with tau = 0 has been considered by Galakhov et al. (2016). This paper mainly deals with the fully parabolic system with tau = 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. (C) 2019 Elsevier Ltd. All rights reserved.