Global solvability and boundedness in the N-dimensional quasilinear chemotaxis model with logistic source and consumption of chemoattractant

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We consider the following chemotaxis model %fully parabolic Keller-Segel system with logistic source {arrayll uₜ=∇·(D(u)∇ u)-χ∇·(u∇ v)+μ(u-u²), x∈ Ω, t>0, vₜ-Δv=-uv, x∈ Ω, t>0, %τwₜ+δw=u, %x∈ Ω, t>0, (∇ D(u)-χu· ∇ v)· ν=∂ v/∂ν=0, x∈ ∂Ω, t>0, u(x,0)=u₀(x), v(x,0)=v₀(x), x∈ Ωarray. on a bounded domain Ωⁿ(N≥1), with smooth boundary ∂Ω, χ and μ are positive constants. Besides appropriate smoothness assumptions, in this paper it is only required that D(u)≥ CD(u+1)ᵐ⁻¹ for all u≥ 0 with some CD > 0 and some m>{arrayll 1-μχ[1+λ₀v₀ₗ∞₍Ω₎2³] if N≤2, % >1+(N+2-2r)+/N+2 if % N+2/2≥ r≥N+2/N, 1 if N≥3, array. then for any sufficiently smooth initial data there exists a classical solution which is global in time and bounded, where λ₀ is a positive constant which is corresponding to the maximal sobolev regularity. The results of this paper extends the results of Jin (J. Diff. Eqns., 263(9)(2017), 5759-5772), who proved the possibility of boundness of weak solutions, in the case m>1 and N=3.