Boundedness and large time behavior for a chemotaxis system with signal-dependent motility and indirect signal consumption

In this paper, we study the following chemotaxis system with signal-dependent motility, indirect signal consumption and logistic source {u(t) =Delta(u gamma(v)) + rho u - mu u(l), x is an element of Omega, t > 0, v(t) = Delta v - vw, x is an element of Omega, t > 0, w(t) = -delta w + u, x is an element of Omega, t > 0 under homogeneous Neumann boundary conditions in a smooth bounded domain Omega subset of R-n, where the motility function gamma(v) is an element of C-3((0, +infinity)), gamma(v) > 0, gamma'(v) < 0 on (0, +infinity), lim(v ->infinity) gamma(v) = 0, rho > 0, mu > 0, l > 1 and delta > 0. The purpose of this paper is to prove that if l > max{1, n/2}, then the system possesses a global solution. In addition, if l satisfies l{>= 2, if n <= 3, > n/2, if n >= 4, then the solution (u, v, w) satisfies parallel to u(., t) - (rho/mu)(1/l-1) parallel to(L infinity(Omega)) +parallel to v(., t)parallel to(L infinity(Omega)) + parallel to w(., t) - 1/delta (rho/mu)(1/l-1)parallel to(L infinity(Omega)) -> 0 as t -> infinity. (C) 2021 Elsevier Ltd. All rights reserved.