Global existence and asymptotic behavior to a chemotaxis system with consumption of chemoattractant in higher dimensions

Abstract

We study the quasilinear chemotaxis system (1.1) in a bounded domain Ω ⊂ ℝn(n ≥ 3) with smooth boundary, where the diffusion function D(u) satisfies D(u) ≥ cDum−1 for all u > 0 with some cD > 0. Under the condition m>32−1n, we show that for all reasonably regular initial data, the corresponding initial-boundary value problem for (1.1) possesses global boundedness of solution, which converges to the spatially homogeneous equilibrium (ū0,0) in an appropriate sense as t → ∞, where ū0=1Ω∫Ωu0.