Boundedness and asymptotic behavior of solutions to a chemotaxis-haptotaxis model in high dimensions

We consider the Neumann value problem for the chemotaxis system {u(t) = del.(del u - u (alpha/1 + v del v + rho del w)) + lambda u(1 - u), x is an element of Omega, t > 0, v(t) = del v - v - mu uv, x is an element of Omega, t > 0, w(t) = gamma u(1 - w), x is an element of Omega, t > 0, in a bounded domain Omega subset of R-n (n >= 1) with smooth boundary, where alpha, rho, lambda, mu, and gamma are positive coefficients. It is shown that for any choice of reasonably regular initial data (u(0), v(0), w(0)), there exists a constant lambda* depending on alpha, rho, mu, gamma, eta, v(0) and wo such that for any lambda > lambda*, the associated initial boundary system possesses a global classical solution which is uniformly bounded. Moreover, building on this boundedness property, it is proved that as time tends to infinity, all the solution approaches the homogeneous steady state (1, 0, 1) in an appropriate sense. (C) 2015 Elsevier Ltd. All rights reserved.