Global classical solutions for a class of reaction-diffusion system with density-suppressed motility

This paper is concerned with a class of reaction-diffusion system with density-suppressed motility <tex-math id="FE1"> {document}$ {equation*} {cases} uₜ = Δ(γ(v) u)+α u F(w), & x ∈ Ω, t>0, \\ vₜ = D Δ v+u-v, & x ∈ Ω, t>0, \\ wₜ = Δ w-u F(w), & x ∈ Ω, t>0, {cases} {equation*} document </tex-math> under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂ R^n\; (n≤ 2) $, where $ α > 0 $ and $ D > 0 $ are constants. The random motility function $ γ satisfies <tex-math id="FE2"> document {equation*} γ∈ C^3((0, +∞)), \ γ>0, \ γ'<0\, \ on\, \ (0, +∞) \ \ and\ \ limv→ +∞γ(v) = 0. {equation*} document </tex-math> The intake rate function F $ satisfies $ F∈ C^1([0, +∞)), \, F(0) = 0\, \ and\ \, F > 0\, \ on\, \ (0, +∞) $. We show that the above system admits a unique global classical solution for all non-negative initial data $ u_0∈ W1, ∞(Ω), \, v_0∈ W1, ∞(Ω), \, w_0∈ W1, ∞(Ω) $. Moreover, if there exist $ k > 0 $ and $ v̄ > 0 such that <tex-math id="FE3"> document {equation*} infv>v̄v^kγ(v)>0, {equation*} ${document} </tex-math> then the global solution is bounded uniformly in time.