A note on global existence to a higher-dimensional quasilinear chemotaxis system with consumption of chemoattractant

The Neumann boundary value problem for the chemotaxis system generalizing the prototype (Equation prsented) 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.