Global boundedness and large time behaviour in a higher-dimensional quasilinear chemotaxis system with consumption of chemoattractant

This paper deals with the following quasilinear chemotaxis system with consumption of chemoattractant \[ \{{array}{@{}ll} u_t=Δ uᵐ-∇·(u∇ v), & x∈ Ω, t>0,\\ v_t=Δ v-uv, & x∈ Ω, t>0\\ {array}. \] in a bounded domain Ω ⊂ RN(N=3,\,4,\,5) with smooth boundary ∂ Ω . It is shown that if m>max \1,\,3N-2/2N+2\ , for any reasonably smooth nonnegative initial data, the corresponding no-flux type initial-boundary value problem possesses a globally bounded weak solution. Furthermore, we prove that the solution converges to the spatially homogeneous equilibrium (ū₀,\,0) in an appropriate sense as t→ ∞ , where ū₀=1/|Ω |∫ _Ω u₀ . This result not only partly extends the previous global boundedness result in Fan and Jin ( J. Math. Phys. 58 (2017), 011503) and Wang and Xiang ( Z. Angew. Math. Phys. 66 (2015), 3159–3179) to m>3N-2/2N in the case N≥ 3 , but also partly improves the global existence result in Zheng and Wang ( Discrete Contin. Dyn. Syst. Ser. B 22 (2017), 669–686) to m>3N/2N+2 when N≥ 2 .