Global boundedness in a quasilinear chemotaxis system with general density-signal governed

In this paper we study the global boundedness of solutions to the quasilinear parabolic chemotaxis system: u(t) = del . (D(u)del u - S(u)del phi(v)), 0 = Delta v - v + u, subject to homogeneous Neumann boundary conditions and the initial data up in a bounded and smooth domain Omega subset of R-n (n >= 2), where the diffusivity D(u) is supposed to satisfy D(u) >= a(0)(u + 1)(-alpha) with a(0) > 0 and alpha is an element of R, while the density -signal governed sensitivity fulfills 0 <= S(u) <= b(0)(u + 1)(beta) and 0 < phi'(v) <= x/v(k) for b(0), x > 0 and beta, k is an element of R. It is shown that the solution is globally bounded if alpha + beta < (1 - 2/n)k + 2/n with n >= 3 and k < 1, or alpha + beta < 1 for k >= 1 This implies that the large k benefits the global boundedness of solutions due to the weaker chemotactic migration of the signal -dependent sensitivity at high signal concentrations. Moreover, when alpha + beta arrives at the critical value, we establish the global boundedness of solutions for the coefficient x properly small. It should be emphasized that the smallness of x under k > 1 is positively related to the total cellular mass integral(Omega)u(0)dx, which is attributed to the stronger singularity of phi(v) at v = 0 for k > 1 and the fact that v can be estimated from below by a multiple of integral(Omega)u(0)dx. In addition, distinctive phenomena concerning this model are observed by comparison with the known results. (C) 2017 Elsevier Inc. All rights reserved.