Global boundedness and large time behavior of a chemotaxis system with indirect signal absorption

The paper considers the following system {u(t) = Delta u - del . (u del v) + f (u) , 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 bounded domain Omega subset of R-n (n >= 2) with smooth boundary, where delta > 0 and f(s) = mu(s - s(r) ) with mu >= 0, r > 1. When mu > 0 we proved for any r > n/2 the system admits a globally bounded classical solution. However, when f (u) = mu(u - u(2) ) with mu > 0 and n >= 4, we establish the global existence and the boundedness of the solution if mu is suitably large. Moreover, by constructing the functional, we proved the solution converges to (1, 0, 1/delta ) in L-infinity (Omega) as t -> infinity under the basic assumptions concerning to the boundedness on mu. Furthermore, by constructing the another functional, we proved the convergence is exponential if mu is suitably large. Finally, we showed the solution of the system with mu = 0 is globally bounded if n >= 3 and 0 < parallel to v(0)parallel to L-infinity (Omega) <= pi/root n, which improves the result of [5]. (c) 2020 Elsevier Inc. All rights reserved.