Global boundedness of solutions to the chemotaxis system with general sensitivity and nonlinear production

In this paper, we consider the following chemotaxis system {equation*} \{ {split} &nₜ=∇·((n+1)ᵐ⁻¹∇ n)-∇·(n∇ v/(1+n)^α),&&x∈ Ω ,t> 0,\\ &0=Δ v-v+n^γ,&&x∈ Ω ,t> 0\\ {split} . {equation*} in a bounded domain ΩN, N≥2 with smooth boundary. It is shown that for all reasonably regular initial data, if α>max\0,γ-m-2/N+1\, γ>0 and m≥1, then the problem possesses a unique global bounded classical solution.