BOUNDEDNESS IN QUASILINEAR KELLER-SEGEL EQUATIONS WITH NONLINEAR SENSITIVITY AND LOGISTIC SOURCE

In this paper, we investigate the quasilinear Keller-Segel equations (q-K-S): {nt = del. (D(n)Vn) - del. (x(n)del c) + R(n), x is an element of Omega, t > 0, rho ct = Delta c - c + n, x is an element of Omega, t > 0, under homogeneous Neumann boundary conditions in a bounded domain Omega subset of R-N. For both rho = 0 (parabolic-elliptic case) and rho > 0 (parabolic-parabolic case), we will show the global-in-time existence and uniform-in-time boundedness of solutions to equations (q-K-S) with both non-degenerate and degenerate diffusions on the non-convex domain Omega, which provide a supplement to the dichotomy boundedness vs. blow-up in parabolic-elliptic/parabolic-parabolic chemotaxis equations with degenerate diffusion, nonlinear sensitivity and logistic source. In particular, we improve the recent results obtained by Wang-LiMu (2014, Disc. Cont. Dyn. Syst.) and Wang-Mu-Zheng (2014, J. Differential Equations).