Large time behavior of solutions to a fully parabolic chemotaxis-haptotaxis model in N dimensions

This paper deals with the Neumann problem for a fully parabolic chemotaxis-haptotaxis model of cancer invasion given by {u(t) = Delta u - chi del . (u del v) - xi del . (u del w) + u(a - mu u(r-1 )- lambda w), x is an element of Omega, t > 0, tau v(t) = Delta v - v + u, x is an element of Omega, t > 0, w(t) = - vw, x is an element of Omega, t > 0. Here, Omega subset of R-N (N >= 1) is a bounded domain with smooth boundary and tau > 0, r > 1, lambda >= 0, a is an element of R, mu, xi and x are positive constants. It is shown that the corresponding initial-boundary value problem possesses a unique global bounded classical solution in the cases r > 2 or r = 2, with mu > mu*= (N-2)+/N(chi + C-beta)C-N/2+1(1/N/2+1) for some positive constants C-beta and CN/2+1. Furthermore, the large time behavior of solutions to the problem is also investigated. Specially speaking, when a is appropriately large, the corresponding solution of the system exponentially decays to ((a/mu)(1/r-1), (a/mu)(1/r-1), 0) if mu is large enough. This result improves or extends previous results of several authors. (C) 2018 Elsevier Inc. All rights reserved.