A NOTE ON GLOBAL EXISTENCE TO A HIGHER-DIMENSIONAL QUASILINEAR CHEMOTAXIS SYSTEM WITH CONSUMPTION OF CHEMOATTRACTANT

摘要

The Neumann boundary value problem for the chemotaxis system generalizing the prototype {u_t = ▽ · (D(u)▽u) - ▽ · (u▽v), x ϵ Ω, t>0, v_t = ∆v - uv, x ϵ Ω,t > 0, (KS) is considered in a smooth bounded convex domain Ω ⊂ R~N(N ≥ 2), where D(u) ≥ C_D(u + 1)~(m-1) for all u ≥ 0 with some m > 1 and C_D > 0. If m > (3N)/(2N+2) and suitable regularity assumptions on the initial data are given, the corresponding initial-boundary problem possesses a global classical solution. Our paper extends the results of Wang et al. ([24]), who showed the global existence of solutions in the cases m > 2 - 6/(N+4) (N ≥ 3). If the flow of fluid is ignored, our result is consistent with and improves the result of Tao, Winkler ([15]) and Tao, Winkler ([17]), who proved the possibility of global boundedness, in the case that N = 2, m > 1 and N = 3, m > 8/7, respectively.