Global existence and boundedness of classical solutions in a quasilinear parabolic-elliptic chemotaxis system with logistic source

We consider the quasilinear parabolic-elliptic chemotaxis system {u(t) = del . (D(u)del u - chi u del v) + g(u), chi is an element of Omega, t > 0, 0 = Delta v - v + u, chi is an element of Omega, t > 0, under homogeneous Neumann boundary conditions in a smooth bounded domain Omega subset of R-n, n >= 1. We assume that the functions D and g are smooth and satisfy D(s) > 0 for s >= 0, D(s) >= C(D)s(m-1) for s > 0, g(0) >= 0, g(s) <= a - bs(gamma), s > 0 with some constants C-D > 0,m >= 1,a >= 0,b > 0 and gamma > 2. We prove that the classical solutions to the above system are uniformly in-time-bounded without any restrictions on m and b. This result extends one of the recent results by Wang et al. (2014) [16], which assert the boundedness of solutions for gamma > 2 under the condition b > b* with b* = 0 for m >= 2 - 2/n and b* = (2-m)n-2/(2-m)n chi for m < 2 - 2/n. (C) 2015 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.